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Wu,1 Y. Dai,1, 2 and M. D. Ding1, 2 +1School of Astronomy and Space Science, Nanjing University, Nanjing 210023, People’s Republic of China +2Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023, People’s Republic +of China +ABSTRACT +Sunquakes are enhanced seismic waves excited in some energetic solar flares. Up to now, their origin +has still been controversial. In this Letter, we select and study 20 strong flares in Solar Cycle 24, +whose impulse phase is fully captured by the Reuven Ramaty High Energy Solar Spectroscopic Imager +(RHESSI ). For 11 out of 12 sunquake-active flares in our sample, the hard X-ray (HXR) emission shows +a good temporal and spatial correlation with the white-light (WL) enhancement and the sunquake. +Spectral analysis also reveals a harder photon spectrum that extends to several hundred keV, implying +a considerable population of flare-accelerated nonthermal electrons at high energies. Quantitatively, +the total energy of electrons above 300 keV in sunquake-active flares is systematically different from +that in sunquake-quiet flares, while the difference is marginal for electrons above 50 keV. All these +facts support highly energetic electrons as a preferred driver of the sunquakes. Such an electron-driven +scenario can be reasonably accommodated in the framework of a recently proposed selection rule for +sunquake generation. For the remaining one event, the sunquake epicenter is cospatial with a magnetic +imprint, i.e., a permanent change of magnetic field on the photosphere. Quantitative calculation shows +that the flare-induced downward Lorentz force can do enough work to power the sunquake, acting as +a viable sunquake driver for this specific event. +Keywords: Solar flares (1496), Solar flare spectra (1982), Solar particle emission (1517), Helioseismol- +ogy (709), Solar x-ray flares (1816), Solar white-light flares (1983) +1. INTRODUCTION +It is believed that solar flares are a result of rapid release of free magnetic energy stored in the solar corona. +Through magnetic reconnection, the magnetic energy is converted to a variety of forms, which are transported both +upward to the interplanetary space and downward to the solar lower atmosphere. +In some energetic flares, the +flare-powered perturbations can reach the dense photosphere to enhance the local helioseismic waves, which further +penetrate through the solar interior and get reflected back to the photosphere, termed as “sunquakes” (Wolff 1972). +The first sunquake observation was reported in Kosovichev & Zharkova (1998), where the wave signature is manifested +as circular “ripples” in Dopplergrams. Since then, more and more such sunquake events have been discovered (e.g., +Donea et al. 1999; Kosovichev 2006; Zharkov et al. 2011). +Up to now, the origin of sunquakes has still been controversial. Several categories of driving mechanisms have been +proposed. The first category assumes flare-accelerated particles as the driver of sunquakes. The sunquakes are excited +either by direct impact of the energetic particles on the photosphere (Kosovichev & Zharkova 1998; Kosovichev 2007; +Zharkova & Zharkov 2007; Kosovichev 2006; Zharkova 2008), or due to pressure pulse from the heated chromosphere +by thick-target bremsstrahlung of the nonthermal electrons (Donea et al. 2006a; Lindsey & Donea 2008). This scenario +is analogous to the mechanism for white-light flares (WLFs) of type I (Hudson 1972; Chen & Ding 2005, 2006), and +is supported by a good correlation between the sunquake source, white-light (WL) enhancement, and hard X-ray +(HXR) emission revealed in many observations (Buitrago-Casas et al. 2015). In another category, it is assumed that a +Corresponding author: Y. Dai +ydai@nju.edu.cn + +2 +Wu et al. +downward Lorentz force resulting from abrupt and permanent changes of the photospheric magnetic field, which often +occur in strong flares (Sudol & Harvey 2005; Petrie & Sudol 2010; Fisher et al. 2012; Sun et al. 2017), can act as a +sunquake driver (Hudson et al. 2008; Fisher et al. 2012). +It has been shown that sunquakes tend to occur in strong flares (Sharykin & Kosovichev 2020). Nevertheless, only +a fraction of strong flares can produce a sunquake. Based on a statistical study of major flares in Solar Cycle 24 +observed by the Solar Dynamics Observatory (SDO; Pesnell et al. 2012) mission, Chen & Zhao (2021, hereafter CZ21) +proposed a selection rule for sunquake generation: a sunquake is more likely to occur when the photosphere shows a +net downward oscillatory velocity. In such a case, the photospheric oscillation can be amplified by the in-phase flare- +excited impulse, facilitating the generation of a sunquake. Otherwise, the background oscillation should be weakened +instead. This may explain the relative rarity of sunquakes in real observations. +The selection role proposed by CZ21 provides a promising explanation for the occurrence rate of sunquakes. However, +the detailed mechanisms for sunquake generation are still poorly understood without resorting to other complementary +observations. In this Letter, we further include HXR imaging and spectroscopic data to the sample sunquakes analyzed +by CZ21, mainly focusing on the possible role of flare-accelerated electrons in producing the sunquakes. +2. INSTRUMENTS AND DATASET +The data used in this study mainly come from the Helioseismic and Magnetic Imager (HMI; Schou et al. 2012) +on board SDO and the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI ; Lin et al. 2002). HMI +measures full-disk Stokes profiles of the Fe I 6173 ˚A line with a pixel size of 0.5′′ and cadence of 45 s, from which data +products such as the continuum intensity (Ic), Doppler velocity, and vector magnetic field of the photosphere can be +derived. RHESSI is designed for imaging and spectroscopic observations of the Sun in X-rays and γ-rays. Using a +rotation modulation of nine detectors with a 4s period, the spacecraft achieves a spatial resolution as high as 2.3′′ and +spectral solution of 1–10 keV over an energy range from 3 keV to 17 MeV. +We start from the sample of events originally compiled in CZ21, which includes the strongest 60 flares in Solar Cycle +24 that occur within 75◦ in longitude. This yields a lower limit of M6.3 in GOES soft X-ray (SXR) class for the +candidate flares. As revealed in the HMI Ic images, all of the flares are strong enough to exhibit a distinguishable WL +emission enhancement, indicative of WLFs with the potential to produce sunquakes. Furthermore, the flare locations +not too close to the limb ensure that the parameters of the possible sunquakes can be credibly derived from the +reconstructed HMI egression power maps. +To investigate the possible role of flare-accelerated electrons in generating sunquakes, we focus on flares whose +impulsive phase is fully captured by RHESSI. We need to apply such an additional selection criterion since RHESSI +observations are routinely affected by orbit night and/or other gaps. Doing so reduces the original sample to 20 flare +events, of which 12 flares are in association with at least one sunquake, while the remaining 8 ones are seismically +quiet. If there are more than one sunquake events in a sunquake-active flare, we consider the most energetic one, +which is usually significantly stronger than the others. The general information of the flares under study, as well as +their characteristics to be quantified in the following analysis, are listed in Table 1. Here the sunquake information +is adopted from CZ21. We note that all but one (associated with the 2011 August 9 X6.9 flare, No. 4) sunquakes in +our list show a net downward oscillatory velocity (in either the 3–5 mHz frequency band or the 5-7 mHz one, or both) +during the flare impulsive phase. +3. ANALYSIS AND RESULTS +Figure 1 depicts the WL and X-ray observations of a typical sunquake-active flare that occurred on 2012 October +23 (No. 7) in NOAA active region 11598. The event has been extensively studied in the literature (e.g., Yang et al. +2015; Sharykin et al. 2017; Watanabe & Imada 2020), and was also selected as a typical example presented in CZ21. +According to the GOES 1–8 ˚A light curve (blue) plotted in Figure 1(a), the SXR flare starts at 03:14 UT, promptly +rises to its peak at 03:17 UT, and ends at 03:21 UT, registered as an X1.8-class flare. The HXR emission of the +flare, as revealed from the RHESSI 50–100 keV count rate (red line in Figure 1(a)), exhibits an even more impulsive +increase and peaks at around 03:16 UT, slightly earlier than the SXR emission, which implies that the “Neupert effect” +(Neupert 1968) applies to this flare. It is also seen that the flare WL emission, which is proxied by the HMI continuum +intensity (black line with triangle symbols in Figure 1(a)) summed over the main flaring region (dashed box in Figure +1(b)), shows a nearly synchronous enhancement with the HXR emission before reaching its maximum at 03:16:15 UT. +After then, the WL emission turns to a relatively gradual decay in comparison with the precipitous drop in HXR +emission. + +ENERGETIC ELECTRONS AS A DRIVER OF SUNQUAKES +3 +As shown in Figure 1(b), the WL enhancement at the peak is predominately manifested as two quasi-parallel flare +ribbons. Here, for clarity of viewing, we subtract a pre-flare image from the image at the flaring time to highlight the +WL enhancement, and plot the base-difference map in an inverse color scale where dark features indicate brightening. +When overplotting a simultaneous RHESSI image at 50–100 keV (red contours) on the HMI WL map, it is seen that +the HXR source well covers the WL ribbons, although the former seems more diffuse. According to Yang et al. (2015), +the WL ribbons correspond to the western segments of a pair of inner/outer circular ribbons that outline the base of +a fan-spine topology, while the HXR source is located around the south footpoint of a magnetic flux rope embedded +under the fan dome. The close temporal and spatial correlation between the WL and HXR emissions indicates that +this event belongs to a type I WL flare, in which the WL emission originates from the layers heated by a direct electron +bombardment and/or the following backwarming effect (Hudson 1972; Chen & Ding 2005, 2006; Hao et al. 2012). +For this sunquake-active flare, we also mark out the location of the sunquake epicenter (green asterisk in Figure 1(b)). +As CZ21 have verified a tight correlation between the WL enhancement and sunquake excitation, our complementary +HXR observations strongly suggest the same electron-driven scenario for the sunquake generation as that for the WL +enhancement in this flare (Sharykin et al. 2017; Watanabe & Imada 2020). By checking other sunquake events, we +find that all but one (the 2011 August 9 X6.9 flare, No. 4) of the sunquakes in our list show a good correlation with the +HXR emission both temporally and spatially, which further corroborates nonthermal electrons as a preferred driver of +the sunquakes. +To further quantify the energetics of flare-accelerated electrons, we fit the RHESSI spectra during the whole flare +impulsive phase (listed in Table 1) using the Object Spectral Executive (OSPEX) package. First, we divide the impul- +sive phase into several time intervals, each of which has a duration of 20 s. Then we use a thick-target bremsstrahlung +model (thick2), which assumes a broken power-law distribution of the flare-accelerated nonthermal electrons, plus a +single-temperature thermal model (vth) to perform the spectral fitting for each individual interval. Since we are only +concerned with nonthermal properties, the thermal component is introduced just to better constrain the low-energy +cutoff (Ec) of the nonthermal electrons. Therefore, the lower limit of the energy range for fitting is fixed at 10 keV to +exclude the Fe/Ni emission lines at ∼6.7 keV, which permits a simplification of the thermal component fitting by only +varying the temperature and emission measure while keeping the elemental abundance unchanged. On the other end, +the upper limit is determined such that the photon flux at that energy starts to drop below the background level. +Figure 2(a) shows the RHESSI spectrum around the HXR peak of the 2012 October 23 flare, as well as the spectral +fitting results. It is seen that the photon flux at 30 keV is as high as 68.9 photon s−1 cm−2 keV−1, among the typical +values observed in WLFs (Kuhar et al. 2016; Hao et al. 2017). More importantly, the flux keeps above the background +level until 400 keV, indicative of a significant fraction of electrons accelerated to very high energies. We note that this +is a common spectral feature for the sunquake-active flares. The spectral fitting reveals power-law indices of 3.96 and +3.42 for the nonthermal electrons below and above a break energy of 461 keV, respectively, reflecting a hardening of +the spectra toward higher energies. +For comparison, we also present in Figures 2(b) and (c) the spectra of the other two flares that are of similar GOES +classes but without sunquakes. For these sunquake-quiet flares, the photon flux at 30 keV is comparable to that for the +sunquake-active events. Toward higher energies, however, the HXR spectrum shows diverse variations either becoming +very soft such that the flux quickly drops below the background (the 2014 October 27 X2.0 flare, No. 18), or still +behaving like that of the sunquake-active events (the 2011 September 24 X1.9 flare, No. 6). Obviously, the diverse +spectral patterns imply that the population of high energy electrons in sunquake-quiet flares can be distinctly different +from case to case. +Based on the spectral fitting, we evaluate the total energy of nonthermal electrons using the integral +E = +�� +εF(ε, t) dεdt, +(1) +where ε is the electron energy and F(ε, t) the fitted electron spectrum. The integration with respect to time is done +over the entire flare impulsive phase. As to the energy range for integration, we adopt fixed lower limits regardless of +the variable low-energy cutoffs derived from actual flares. Here we calculate the total energies of the electrons above 50 +keV (E50) and that above 300 keV (E300), which characterize the energetics of mildly and highly energetic electrons, +respectively. +Figure 3 displays the histograms of E50 (left) and E300 (right) for the flares with (upper) and without (lower) +sunquakes, respectively. Note that we exclude the 2011 August 9 sunquake-active flare in which the sunquake originates +in a different place from that for the nonthermal electrons. It is found that the distribution of E50 for sunquake-active + +4 +Wu et al. +flares shows no significant difference from that for sunquake-quiet flares; both distributions span over a similar energy +range and peak at 1029.5–1030 erg (Figures 3(a) and (b)). Nevertheless, a systematic difference is seen in the distribution +of E300. The E300 value for the flares with sunquakes varies in a relatively narrow range, and is dominantly restricted +to a magnitude of 1027–1028 erg (Figure 3(c)), which is comparable to the estimated energy of sunquakes reported in +previous studies (Donea et al. 2006b; Chen & Zhao 2021). By contrast, the value of E300 for the sunquake-quiet flares +seems more scattered, which is either comparable to that for the sunquake-active flares, or several orders of magnitude +lower (Figure 3(d)). Such a bimodal distribution can be expected from the spectral fitting for the sunquake-quiet flares +shown in Figure 2. +We also calculate the corresponding electron power, which is obtained by dividing the total electron energy by the +duration of impulsive phase. As shown in Table 1, the length of impulsive phase just varies in a narrow range of 60–120 +s from event to event. It is found that the distributions of the electron power (not shown here) are nearly the same as +those shown in Figure 3. +The above statistical result implies that the generation of the sunquakes is more relevant to highly energetic electrons +rather than electrons at moderate energies. However, the latter is more likely to be responsible for the enhancement of +WL emission. Furthermore, the electron-driven scenario for sunquakes can be reasonably accommodated in the frame +of the selection rule proposed by CZ21. In addition to being in phase with the background oscillation, the downward +electron beam should contain enough highly accelerated electrons in order to efficiently perturb the photosphere and +deep layers to produce a sunquake. As for the sunquake-quiet flares, however, either the electron-driven impulse is too +weak (e.g., the 2014 October 27 X2.0 flare shown in Figure 2(b)), or the impulse is out of phase with the background +oscillation (e.g., the 2011 September 24 X1.9 flare flare shown in Figure 2(c)), thus unable to generate a sunquake. +This is also the reason why the distribution of E300 is more scattered for the flares without sunquakes. +Among all the sunquake-active events, the 2011 August 9 flare is an exception in that its sunquake epicenter is +spatially offset with the HXR source, which requires an alternative explanation for the sunquake generation. Previous +observations have shown that some major solar flares can leave magnetic imprints (MIs) on the photosphere, which are +manifested as rapid and irreversible changes of the photospheric magnetic field (Lu et al. 2019). During this process, +the photospheric magnetic field becomes more horizontal, producing a downward Lorentz force on the photosphere +that possibly drives a sunquake (Hudson et al. 2008). In the following, we test the possibility of flare-induced Lorentz +force as the sunquake driver for this specific event. +To depict the MIs accurately, we use Space-weather HMI Active Region Patch (SHARP; Bobra et al. 2014) products, +whose data pipeline includes a remapping of the magnetic field vector in a cylindrical equal-area (CEA) projection. +The three components of the SHARP magnetic field vector are represented by Br (radial), Bp (southward), and Bt +(westward), respectively, from which the magnitude of the horizontal magnetic field is derived as Bh = +� +B2p + B2 +t . +Since the flare-induced magnetic field change is mainly reflected in an increase of the horizontal magnetic field, we use +regions where δBh exceeds a threshold (e.g., 300 G) to approximate the spatial extent of MIs (cf. Lu et al. 2019). +We plot in Figure 4(a) the locations of the MIs (orange plus yellow contours), HXR source (red contours), and +sunquake epicenter (green asterisk) for the 2011 August 9 flare, which are overlaid on the corresponding HMI continuum +map. As shown in the figure, the MIs appear patch-like, and are located predominately in the vicinity of or over the +polarity inversion line (PIL) of SHARP Br, consistent with many previous observations (e.g., Petrie 2012, 2013; +Wang et al. 2012a,b; Sun et al. 2012). The sunquake epicenter lies exactly in a southern MI (distinguished with the +other MIs in yellow contours) but distant from the HXR source, which does suggest a Lorentz force-driven origin of +the sunquake. +Compared with other MIs, the sunquake-related MI is located in an isolated region near the far end of the PIL, +where the background magnetic field is relatively weaker than that in the AR core. In addition, it appears neither +too diffuse nor too compact. These facts may reflect necessary physical conditions for an MI to generate sunquakes. +Nevertheless, without other observations of such MI-related sunquakes our argument is not conclusive. +Quantitatively, we use the equation +δF = 1 +8π +� +Aph +(δB2 +r − δB2 +h) dA +(2) +to calculate the Lorentz force δF over this sunquake-related MI (Hudson et al. 2008). When considering an MI area +of Aph = 1.3 × 1017 cm2 surrounding the sunquake epicenter if we select a threshold of δBh = 300 G (enclosed by the +outermost yellow contour), the resultant downward Lorentz force on this area is 1.2 × 1022 dyne. By further assuming +a displacement of 3 km that the Lorentz force pushes the photosphere downward (cf. Hudson et al. 2008), we derive + +ENERGETIC ELECTRONS AS A DRIVER OF SUNQUAKES +5 +a work of 3.8 × 1027 erg done by the Lorentz force, which is close to the sunquake energy for this event estimated in +CZ21. Compared with the impulsive perturbation by energetic electrons, the MI-induced Lorentz force should act on +the photosphere in a much more gentle manner. We note that this sunquake event presents a nearly zero net oscillatory +velocity in contrast to the other events. +Finally we present the corresponding observations of the 2012 October 23 event in Figure 4(b) for comparison. +Although the MIs in this flare still gather along the PIL, the sunquake epicenter shows an offset with respect to the +MIs in spite of a significant line-shortening due to a close-to-the-limb location of the flare. Instead, the sunquake site +should be located in the inner circular flare ribbon. +4. DISCUSSION AND CONCLUSION +In this Letter, we make a statistical study on sunquake generation using a sample of 20 strong solar flares that have a +full RHESSI coverage of the impulsive phase. For 11 out of 12 sunquake-active flares in our sample, the HXR emission +shows a good temporal and spatial correlation with the WL enhancement and the sunquake. Spectral analysis also +reveals a hard photon spectrum in which the photon flux is well above the background level until several hundred keV, +implying a significant population of flare-accelerated nonthermal electrons at high energies. Furthermore, the total +energies of electrons above 300 keV in sunquake-active flares are systematically different from those of sunquake-quiet +flares, while the difference is marginal for energies above 50 keV. All these facts support highly energetic electrons as a +preferred driver of the sunquakes. Besides the selection rule proposed in CZ21, i.e., the flare-induced impulsive heating +should be in phase with a downward background oscillation, a strong electron beam with in particular a significant +fraction of energy residing in highly energetic electrons should serve as another necessary condition for the sunquake +generation. If either of the two conditions is broken down, a sunquake is not likely to occur. +According to Neidig (1989), only electrons above an energy of ∼900 keV can penetrate to the photosphere. Nev- +ertheless, in a flaring atmosphere, the ionization, condensation, and evaporation of plasma may mitigate the energy +requirement for the electrons to reach such depths (Watanabe & Imada 2020). In this meaning, the electron-driven +sunquakes in our sample could be excited by the direct impact of extremely energetic electrons on the photosphere +(Kosovichev & Zharkova 1998; Kosovichev 2007; Zharkova & Zharkov 2007; Kosovichev 2006; Zharkova 2008). Nev- +ertheless, it is also possible that the pressure pulse from the heated chromosphere by less energetic electrons plays a +part role(Donea et al. 2006a; Lindsey & Donea 2008). Without sophisticated radiative hydrodynamic modeling, we +do not intend to clarify the quantitative contributions of these mechanisms for the sunquake generation, which should +be case-dependent. +There is also an exceptional event (the 2011 August 9 sunquake) in our sample, whose sunquake epicenter is cospatial +with an MI instead of the HXR source. We calculate the Lorentz force due to a permanent change of the photospheric +magnetic field over this MI, and estimate the work done by the downward Lorentz force. The quantitative analysis +shows that the magnetic reconfiguration can provide enough energy to power the sunquake. Therefore, although we +suggest highly energetic electrons as a main driver of sunquakes, we do not rule out the role of flare-induced Lorentz +force in some specific events (Hudson et al. 2008; Fisher et al. 2012). +The properties (location and oscillatory velocity) of the electron-driven sunquakes seem different from those of the +MI-related sunquake. Actually, we have checked all electron-driven sunquake events in Table 1, none of which shows +a spatial correspondence with an MI region. Whether it is of physical significance or just a coincidence, we need more +observations to address this issue. +This study only covers a sample of 20 events satisfying our selection criteria that the RHESSI era can provide. In +order to reach a more conclusive result, more events are required. RHESSI has been decommissioned since 2018. +Fortunately, we can make use of imaging and spectroscopic observations with the Spectrometer/Telescope for Imaging +X-rays (STIX) on board the newly launched Solar Orbiter (SolO) mission (Krucker et al. 2020) and the Hard X-ray +Imager (HXI) on board the upcoming Advanced Space-based Solar Observatory (ASO-S) emission (Zhang et al. 2019). +These new observational facilities will help us better understand the origin of sunquakes. +We are grateful to the anonymous referee for his/her insightful comments and suggestions, which led to a signifi- +cant improvement of the manuscript. This work was supported by National Natural Science Foundation of China +under grants 11733003 and 12127901. Y.D. is also sponsored by National Key R&D Program of China under grants +2019YFA0706601 and 2020YFC2201201. SDO is a mission of NASA’s Living With a Star (LWS) program. + +6 +Wu et al. +REFERENCES +Bobra, M. G., Sun, X., Hoeksema, J. 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List of the Flares under study and the Sunquake Information +No. +Date +GOES +RHESSI HXR Information +HMI Sunquake Information +Class +Impulsive Phase +Peaka +E50 +E300 +Sunquake +Correlation +v35b +v57b +(UT) +(UT) +(1030 erg) +(1027 erg) +(Y/N) +(HXR/MI) +(m s−1) +(m s−1) +1 +2011 Feb 13 +M6.6 +17:33:28–17:34:48 +17:34:18 +0.1 +0.02 +N +2 +2011 Feb 15 +X2.2 +01:54:24–01:56:04 +01:55:14 +0.4 +1.4 +Y +HXR +27 +29 +3 +2011 Jul 30 +M9.3 +02:07:28–02:08:48 +02:08:18 +0.2 +0.3 +Y +HXR +417 +337 +4 +2011 Aug 9 +X6.9 +08:02:40–08:04:20 +08:03:50 +3.2 +8.9 +Y +MIc +· · · +-3 +5 +2011 Sep 6 +X2.1 +22:18:20–22:19:40 +22:19:10 +0.8 +29.3 +Y +HXR +326 +596 +6 +2011 Sep 24 +X1.9 +09:35:16–09:36:56 +09:36:26 +0.5 +22.6 +N +7 +2012 Oct 23 +X1.8 +03:15:08–03:16:08 +03:15:58 +1.1 +23.1 +Y +HXR +1082 +950 +8 +2013 May 15 +X1.2 +01:41:20–01:43:00 +01:42:10 +0.4 +4.7 +N +9 +2013 Oct 25 +X1.7 +07:58:10–07:59:50 +07:59:20 +0.6 +9.2 +Y +HXR +· · · +135 +10 +2013 Oct 25 +X2.1 +15:00:12–15:01:52 +15:00:42 +0.6 +5.4 +N +11 +2013 Oct 28 +X1.0 +01:58:48–02:00:28 +01:59:38 +0.3 +10.6 +N +12 +2013 Nov 10 +X1.1 +05:12:10–05:13:50 +05:12:40 +0.2 +2.7 +Y +HXR +445 +508 +13 +2014 Jan 7 +M7.2 +10:10:48–10:12:28 +10:11:38 +0.5 +5.6 +Y +HXR +436 +680 +14 +2014 Mar 29 +X1.0 +17:46:00–17:47:40 +17:46:30 +0.2 +11.0 +N +15 +2014 Jun 11 +X1.0 +09:04:20–09:05:40 +09:04:50 +0.06 +5.6 +Y +HXR +· · · +1338 +16 +2014 Oct 22 +M8.7 +01:38:36–01:40:16 +01:39:26 +0.3 +1.0 +Y +HXR +133 +-1 +17 +2014 Oct 22 +X1.6 +14:05:00–14:06:40 +14:06:30 +3.9 +1.4 +Y +HXR +96 +-41 +18 +2014 Oct 27 +X2.0 +14:21:20–14:23:20 +14:23:10 +1.4 +0.04 +N +19 +2015 Mar 7 +M9.2 +22:03:40–22:05:00 +22:04:30 +0.01 +1.4e-5 +N +20 +2017 Sep 7 +M7.3 +10:14:28–10:16:08 +10:15:38 +0.4 +8.8 +Y +HXR +534 +344 +Note— +a Peak time for HXR emission at 50–100 keV. +b Oscillatory velocities at 3–5 MHz (v35) and 5–7 MHz (v57), respectively. The values are adopted from CZ21. +c The estimated work done by the MI-induced Lorentz force is 3.8 × 1027 erg. + +8 +Wu et al. +Figure 1. WL and X-ray observations of the 2012 October 23 X1.8 flare. (a) time profiles of the HMI continuum intensity +around 6173 ˚A (black), RHESSI HXR count rate at 50–100 keV (red), and GOES SXR flux in 1–8 ˚A (blue). (b) the base- +difference HMI continuum map at the continuum peak time in an inverse scale, where the dashed box encloses the main flaring +region used for continuum calculation. Overplotted on the map is a simultaneous RHESSI 50–100 keV image reconstructed +using the Pixon algorithm, with contour levels corresponding to 30%, 60%, and 90% of the maximum intensity, respectively. +For this sunquake-active flare, the location of the sunquake epicenter is also marked out with an asterisk sign. + +ENERGETIC ELECTRONS AS A DRIVER OF SUNQUAKES +9 +Figure 2. Fitting results for the RHESSI spectra taken around the HXR peak in three flares. In each panel, the event number +in Table 1 is labeled in the upper left, the black and grey lines in histogram mode denote the background-subtracted photon +flux and the background, respectively, while the colored curves represent different components of the modeled spectrum based +on the best-fit parameters. In addition, the residual between the modeled and observed spectra is plotted in the bottom part of +each panel. + +10 +Wu et al. +Figure 3. +Histograms of the total energy of nonthermal electrons for the flare events with (upper) and without (lower) +sunquakes. The left panels are for the distributions of E50 while the right for E300. Note that a bin size of 0.5 dex is adopted +for the histogram plotting. + +Figure 4. Locations of the MIs (orange plus yellow contours), HXR source (red contours), and sunquake epicenter (green +asterisk) for two flares. In each panel the background image is a corresponding HMI continuum map, with the PIL drawn in +blue line. The contours for MI indicate an increase of the horizontal magnetic field at levels of 300 G and 600 G, respectively. +Note that the sunquake-related MI in panel (a) is highlighted in yellow contours. + diff --git a/-NE1T4oBgHgl3EQfCgJv/content/tmp_files/load_file.txt b/-NE1T4oBgHgl3EQfCgJv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0ad95f8b77810b41468a18dec194163390dd99b --- /dev/null +++ b/-NE1T4oBgHgl3EQfCgJv/content/tmp_files/load_file.txt @@ -0,0 +1,593 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf,len=592 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='02865v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='SR] 7 Jan 2023 Draft version January 10, 2023 Typeset using LATEX default style in AASTeX631 Highly Energetic Electrons Accelerated in Strong Solar Flares as a Preferred Driver of Sunquakes H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Wu,1 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Dai,1, 2 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Ding1, 2 1School of Astronomy and Space Science, Nanjing University, Nanjing 210023, People’s Republic of China 2Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023, People’s Republic of China ABSTRACT Sunquakes are enhanced seismic waves excited in some energetic solar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Up to now, their origin has still been controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In this Letter, we select and study 20 strong flares in Solar Cycle 24, whose impulse phase is fully captured by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' For 11 out of 12 sunquake-active flares in our sample, the hard X-ray (HXR) emission shows a good temporal and spatial correlation with the white-light (WL) enhancement and the sunquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Spectral analysis also reveals a harder photon spectrum that extends to several hundred keV, implying a considerable population of flare-accelerated nonthermal electrons at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Quantitatively, the total energy of electrons above 300 keV in sunquake-active flares is systematically different from that in sunquake-quiet flares, while the difference is marginal for electrons above 50 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' All these facts support highly energetic electrons as a preferred driver of the sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Such an electron-driven scenario can be reasonably accommodated in the framework of a recently proposed selection rule for sunquake generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' For the remaining one event, the sunquake epicenter is cospatial with a magnetic imprint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=', a permanent change of magnetic field on the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Quantitative calculation shows that the flare-induced downward Lorentz force can do enough work to power the sunquake, acting as a viable sunquake driver for this specific event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Keywords: Solar flares (1496), Solar flare spectra (1982), Solar particle emission (1517), Helioseismol- ogy (709), Solar x-ray flares (1816), Solar white-light flares (1983) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' INTRODUCTION It is believed that solar flares are a result of rapid release of free magnetic energy stored in the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Through magnetic reconnection, the magnetic energy is converted to a variety of forms, which are transported both upward to the interplanetary space and downward to the solar lower atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In some energetic flares, the flare-powered perturbations can reach the dense photosphere to enhance the local helioseismic waves, which further penetrate through the solar interior and get reflected back to the photosphere, termed as “sunquakes” (Wolff 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The first sunquake observation was reported in Kosovichev & Zharkova (1998), where the wave signature is manifested as circular “ripples” in Dopplergrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Since then, more and more such sunquake events have been discovered (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=', Donea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Kosovichev 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Zharkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Up to now, the origin of sunquakes has still been controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Several categories of driving mechanisms have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The first category assumes flare-accelerated particles as the driver of sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The sunquakes are excited either by direct impact of the energetic particles on the photosphere (Kosovichev & Zharkova 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Kosovichev 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Zharkova & Zharkov 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Kosovichev 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Zharkova 2008), or due to pressure pulse from the heated chromosphere by thick-target bremsstrahlung of the nonthermal electrons (Donea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2006a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Lindsey & Donea 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' This scenario is analogous to the mechanism for white-light flares (WLFs) of type I (Hudson 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Chen & Ding 2005, 2006), and is supported by a good correlation between the sunquake source, white-light (WL) enhancement, and hard X-ray (HXR) emission revealed in many observations (Buitrago-Casas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In another category, it is assumed that a Corresponding author: Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Dai ydai@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='cn 2 Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' downward Lorentz force resulting from abrupt and permanent changes of the photospheric magnetic field, which often occur in strong flares (Sudol & Harvey 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Petrie & Sudol 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2017), can act as a sunquake driver (Hudson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' It has been shown that sunquakes tend to occur in strong flares (Sharykin & Kosovichev 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Nevertheless, only a fraction of strong flares can produce a sunquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Based on a statistical study of major flares in Solar Cycle 24 observed by the Solar Dynamics Observatory (SDO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Pesnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2012) mission, Chen & Zhao (2021, hereafter CZ21) proposed a selection rule for sunquake generation: a sunquake is more likely to occur when the photosphere shows a net downward oscillatory velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In such a case, the photospheric oscillation can be amplified by the in-phase flare- excited impulse, facilitating the generation of a sunquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Otherwise, the background oscillation should be weakened instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' This may explain the relative rarity of sunquakes in real observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The selection role proposed by CZ21 provides a promising explanation for the occurrence rate of sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' However, the detailed mechanisms for sunquake generation are still poorly understood without resorting to other complementary observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In this Letter, we further include HXR imaging and spectroscopic data to the sample sunquakes analyzed by CZ21, mainly focusing on the possible role of flare-accelerated electrons in producing the sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' INSTRUMENTS AND DATASET The data used in this study mainly come from the Helioseismic and Magnetic Imager (HMI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Schou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2012) on board SDO and the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' HMI measures full-disk Stokes profiles of the Fe I 6173 ˚A line with a pixel size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='5′′ and cadence of 45 s, from which data products such as the continuum intensity (Ic), Doppler velocity, and vector magnetic field of the photosphere can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' RHESSI is designed for imaging and spectroscopic observations of the Sun in X-rays and γ-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Using a rotation modulation of nine detectors with a 4s period, the spacecraft achieves a spatial resolution as high as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='3′′ and spectral solution of 1–10 keV over an energy range from 3 keV to 17 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' We start from the sample of events originally compiled in CZ21, which includes the strongest 60 flares in Solar Cycle 24 that occur within 75◦ in longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' This yields a lower limit of M6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='3 in GOES soft X-ray (SXR) class for the candidate flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' As revealed in the HMI Ic images, all of the flares are strong enough to exhibit a distinguishable WL emission enhancement, indicative of WLFs with the potential to produce sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Furthermore, the flare locations not too close to the limb ensure that the parameters of the possible sunquakes can be credibly derived from the reconstructed HMI egression power maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' To investigate the possible role of flare-accelerated electrons in generating sunquakes, we focus on flares whose impulsive phase is fully captured by RHESSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' We need to apply such an additional selection criterion since RHESSI observations are routinely affected by orbit night and/or other gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Doing so reduces the original sample to 20 flare events, of which 12 flares are in association with at least one sunquake, while the remaining 8 ones are seismically quiet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' If there are more than one sunquake events in a sunquake-active flare, we consider the most energetic one, which is usually significantly stronger than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The general information of the flares under study, as well as their characteristics to be quantified in the following analysis, are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Here the sunquake information is adopted from CZ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' We note that all but one (associated with the 2011 August 9 X6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='9 flare, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 4) sunquakes in our list show a net downward oscillatory velocity (in either the 3–5 mHz frequency band or the 5-7 mHz one, or both) during the flare impulsive phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' ANALYSIS AND RESULTS Figure 1 depicts the WL and X-ray observations of a typical sunquake-active flare that occurred on 2012 October 23 (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 7) in NOAA active region 11598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The event has been extensively studied in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=', Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Sharykin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Watanabe & Imada 2020), and was also selected as a typical example presented in CZ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' According to the GOES 1–8 ˚A light curve (blue) plotted in Figure 1(a), the SXR flare starts at 03:14 UT, promptly rises to its peak at 03:17 UT, and ends at 03:21 UT, registered as an X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='8-class flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The HXR emission of the flare, as revealed from the RHESSI 50–100 keV count rate (red line in Figure 1(a)), exhibits an even more impulsive increase and peaks at around 03:16 UT, slightly earlier than the SXR emission, which implies that the “Neupert effect” (Neupert 1968) applies to this flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' It is also seen that the flare WL emission, which is proxied by the HMI continuum intensity (black line with triangle symbols in Figure 1(a)) summed over the main flaring region (dashed box in Figure 1(b)), shows a nearly synchronous enhancement with the HXR emission before reaching its maximum at 03:16:15 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' After then, the WL emission turns to a relatively gradual decay in comparison with the precipitous drop in HXR emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' ENERGETIC ELECTRONS AS A DRIVER OF SUNQUAKES 3 As shown in Figure 1(b), the WL enhancement at the peak is predominately manifested as two quasi-parallel flare ribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Here, for clarity of viewing, we subtract a pre-flare image from the image at the flaring time to highlight the WL enhancement, and plot the base-difference map in an inverse color scale where dark features indicate brightening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' When overplotting a simultaneous RHESSI image at 50–100 keV (red contours) on the HMI WL map, it is seen that the HXR source well covers the WL ribbons, although the former seems more diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' According to Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' (2015), the WL ribbons correspond to the western segments of a pair of inner/outer circular ribbons that outline the base of a fan-spine topology, while the HXR source is located around the south footpoint of a magnetic flux rope embedded under the fan dome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The close temporal and spatial correlation between the WL and HXR emissions indicates that this event belongs to a type I WL flare, in which the WL emission originates from the layers heated by a direct electron bombardment and/or the following backwarming effect (Hudson 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Chen & Ding 2005, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' For this sunquake-active flare, we also mark out the location of the sunquake epicenter (green asterisk in Figure 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' As CZ21 have verified a tight correlation between the WL enhancement and sunquake excitation, our complementary HXR observations strongly suggest the same electron-driven scenario for the sunquake generation as that for the WL enhancement in this flare (Sharykin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Watanabe & Imada 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' By checking other sunquake events, we find that all but one (the 2011 August 9 X6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='9 flare, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 4) of the sunquakes in our list show a good correlation with the HXR emission both temporally and spatially, which further corroborates nonthermal electrons as a preferred driver of the sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' To further quantify the energetics of flare-accelerated electrons, we fit the RHESSI spectra during the whole flare impulsive phase (listed in Table 1) using the Object Spectral Executive (OSPEX) package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' First, we divide the impul- sive phase into several time intervals, each of which has a duration of 20 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Then we use a thick-target bremsstrahlung model (thick2), which assumes a broken power-law distribution of the flare-accelerated nonthermal electrons, plus a single-temperature thermal model (vth) to perform the spectral fitting for each individual interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Since we are only concerned with nonthermal properties, the thermal component is introduced just to better constrain the low-energy cutoff (Ec) of the nonthermal electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Therefore, the lower limit of the energy range for fitting is fixed at 10 keV to exclude the Fe/Ni emission lines at ∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='7 keV, which permits a simplification of the thermal component fitting by only varying the temperature and emission measure while keeping the elemental abundance unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' On the other end, the upper limit is determined such that the photon flux at that energy starts to drop below the background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Figure 2(a) shows the RHESSI spectrum around the HXR peak of the 2012 October 23 flare, as well as the spectral fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' It is seen that the photon flux at 30 keV is as high as 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='9 photon s−1 cm−2 keV−1, among the typical values observed in WLFs (Kuhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' More importantly, the flux keeps above the background level until 400 keV, indicative of a significant fraction of electrons accelerated to very high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' We note that this is a common spectral feature for the sunquake-active flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The spectral fitting reveals power-law indices of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='96 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='42 for the nonthermal electrons below and above a break energy of 461 keV, respectively, reflecting a hardening of the spectra toward higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' For comparison, we also present in Figures 2(b) and (c) the spectra of the other two flares that are of similar GOES classes but without sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' For these sunquake-quiet flares, the photon flux at 30 keV is comparable to that for the sunquake-active events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Toward higher energies, however, the HXR spectrum shows diverse variations either becoming very soft such that the flux quickly drops below the background (the 2014 October 27 X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='0 flare, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 18), or still behaving like that of the sunquake-active events (the 2011 September 24 X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='9 flare, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Obviously, the diverse spectral patterns imply that the population of high energy electrons in sunquake-quiet flares can be distinctly different from case to case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Based on the spectral fitting, we evaluate the total energy of nonthermal electrons using the integral E = �� εF(ε, t) dεdt, (1) where ε is the electron energy and F(ε, t) the fitted electron spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The integration with respect to time is done over the entire flare impulsive phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' As to the energy range for integration, we adopt fixed lower limits regardless of the variable low-energy cutoffs derived from actual flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Here we calculate the total energies of the electrons above 50 keV (E50) and that above 300 keV (E300), which characterize the energetics of mildly and highly energetic electrons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Figure 3 displays the histograms of E50 (left) and E300 (right) for the flares with (upper) and without (lower) sunquakes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Note that we exclude the 2011 August 9 sunquake-active flare in which the sunquake originates in a different place from that for the nonthermal electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' It is found that the distribution of E50 for sunquake-active 4 Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' flares shows no significant difference from that for sunquake-quiet flares;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' both distributions span over a similar energy range and peak at 1029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='5–1030 erg (Figures 3(a) and (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Nevertheless, a systematic difference is seen in the distribution of E300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The E300 value for the flares with sunquakes varies in a relatively narrow range, and is dominantly restricted to a magnitude of 1027–1028 erg (Figure 3(c)), which is comparable to the estimated energy of sunquakes reported in previous studies (Donea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2006b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Chen & Zhao 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' By contrast, the value of E300 for the sunquake-quiet flares seems more scattered, which is either comparable to that for the sunquake-active flares, or several orders of magnitude lower (Figure 3(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Such a bimodal distribution can be expected from the spectral fitting for the sunquake-quiet flares shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' We also calculate the corresponding electron power, which is obtained by dividing the total electron energy by the duration of impulsive phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' As shown in Table 1, the length of impulsive phase just varies in a narrow range of 60–120 s from event to event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' It is found that the distributions of the electron power (not shown here) are nearly the same as those shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The above statistical result implies that the generation of the sunquakes is more relevant to highly energetic electrons rather than electrons at moderate energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' However, the latter is more likely to be responsible for the enhancement of WL emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Furthermore, the electron-driven scenario for sunquakes can be reasonably accommodated in the frame of the selection rule proposed by CZ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In addition to being in phase with the background oscillation, the downward electron beam should contain enough highly accelerated electrons in order to efficiently perturb the photosphere and deep layers to produce a sunquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' As for the sunquake-quiet flares, however, either the electron-driven impulse is too weak (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=', the 2014 October 27 X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='0 flare shown in Figure 2(b)), or the impulse is out of phase with the background oscillation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=', the 2011 September 24 X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='9 flare flare shown in Figure 2(c)), thus unable to generate a sunquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' This is also the reason why the distribution of E300 is more scattered for the flares without sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Among all the sunquake-active events, the 2011 August 9 flare is an exception in that its sunquake epicenter is spatially offset with the HXR source, which requires an alternative explanation for the sunquake generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Previous observations have shown that some major solar flares can leave magnetic imprints (MIs) on the photosphere, which are manifested as rapid and irreversible changes of the photospheric magnetic field (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' During this process, the photospheric magnetic field becomes more horizontal, producing a downward Lorentz force on the photosphere that possibly drives a sunquake (Hudson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In the following, we test the possibility of flare-induced Lorentz force as the sunquake driver for this specific event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' To depict the MIs accurately, we use Space-weather HMI Active Region Patch (SHARP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Bobra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2014) products, whose data pipeline includes a remapping of the magnetic field vector in a cylindrical equal-area (CEA) projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The three components of the SHARP magnetic field vector are represented by Br (radial), Bp (southward), and Bt (westward), respectively, from which the magnitude of the horizontal magnetic field is derived as Bh = � B2p + B2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Since the flare-induced magnetic field change is mainly reflected in an increase of the horizontal magnetic field, we use regions where δBh exceeds a threshold (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=', 300 G) to approximate the spatial extent of MIs (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' We plot in Figure 4(a) the locations of the MIs (orange plus yellow contours), HXR source (red contours), and sunquake epicenter (green asterisk) for the 2011 August 9 flare, which are overlaid on the corresponding HMI continuum map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' As shown in the figure, the MIs appear patch-like, and are located predominately in the vicinity of or over the polarity inversion line (PIL) of SHARP Br, consistent with many previous observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=', Petrie 2012, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2012a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The sunquake epicenter lies exactly in a southern MI (distinguished with the other MIs in yellow contours) but distant from the HXR source, which does suggest a Lorentz force-driven origin of the sunquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Compared with other MIs, the sunquake-related MI is located in an isolated region near the far end of the PIL, where the background magnetic field is relatively weaker than that in the AR core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In addition, it appears neither too diffuse nor too compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' These facts may reflect necessary physical conditions for an MI to generate sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Nevertheless, without other observations of such MI-related sunquakes our argument is not conclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Quantitatively, we use the equation δF = 1 8π � Aph (δB2 r − δB2 h) dA (2) to calculate the Lorentz force δF over this sunquake-related MI (Hudson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' When considering an MI area of Aph = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='3 × 1017 cm2 surrounding the sunquake epicenter if we select a threshold of δBh = 300 G (enclosed by the outermost yellow contour), the resultant downward Lorentz force on this area is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='2 × 1022 dyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' By further assuming a displacement of 3 km that the Lorentz force pushes the photosphere downward (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Hudson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2008), we derive ENERGETIC ELECTRONS AS A DRIVER OF SUNQUAKES 5 a work of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='8 × 1027 erg done by the Lorentz force, which is close to the sunquake energy for this event estimated in CZ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Compared with the impulsive perturbation by energetic electrons, the MI-induced Lorentz force should act on the photosphere in a much more gentle manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' We note that this sunquake event presents a nearly zero net oscillatory velocity in contrast to the other events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Finally we present the corresponding observations of the 2012 October 23 event in Figure 4(b) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Although the MIs in this flare still gather along the PIL, the sunquake epicenter shows an offset with respect to the MIs in spite of a significant line-shortening due to a close-to-the-limb location of the flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Instead, the sunquake site should be located in the inner circular flare ribbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION In this Letter, we make a statistical study on sunquake generation using a sample of 20 strong solar flares that have a full RHESSI coverage of the impulsive phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' For 11 out of 12 sunquake-active flares in our sample, the HXR emission shows a good temporal and spatial correlation with the WL enhancement and the sunquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Spectral analysis also reveals a hard photon spectrum in which the photon flux is well above the background level until several hundred keV, implying a significant population of flare-accelerated nonthermal electrons at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Furthermore, the total energies of electrons above 300 keV in sunquake-active flares are systematically different from those of sunquake-quiet flares, while the difference is marginal for energies above 50 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' All these facts support highly energetic electrons as a preferred driver of the sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Besides the selection rule proposed in CZ21, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=', the flare-induced impulsive heating should be in phase with a downward background oscillation, a strong electron beam with in particular a significant fraction of energy residing in highly energetic electrons should serve as another necessary condition for the sunquake generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' If either of the two conditions is broken down, a sunquake is not likely to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' According to Neidig (1989), only electrons above an energy of ∼900 keV can penetrate to the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Nev- ertheless, in a flaring atmosphere, the ionization, condensation, and evaporation of plasma may mitigate the energy requirement for the electrons to reach such depths (Watanabe & Imada 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In this meaning, the electron-driven sunquakes in our sample could be excited by the direct impact of extremely energetic electrons on the photosphere (Kosovichev & Zharkova 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Kosovichev 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Zharkova & Zharkov 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Kosovichev 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Zharkova 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Nev- ertheless, it is also possible that the pressure pulse from the heated chromosphere by less energetic electrons plays a part role(Donea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2006a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Lindsey & Donea 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Without sophisticated radiative hydrodynamic modeling, we do not intend to clarify the quantitative contributions of these mechanisms for the sunquake generation, which should be case-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' There is also an exceptional event (the 2011 August 9 sunquake) in our sample, whose sunquake epicenter is cospatial with an MI instead of the HXR source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' We calculate the Lorentz force due to a permanent change of the photospheric magnetic field over this MI, and estimate the work done by the downward Lorentz force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The quantitative analysis shows that the magnetic reconfiguration can provide enough energy to power the sunquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Therefore, although we suggest highly energetic electrons as a main driver of sunquakes, we do not rule out the role of flare-induced Lorentz force in some specific events (Hudson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The properties (location and oscillatory velocity) of the electron-driven sunquakes seem different from those of the MI-related sunquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Actually, we have checked all electron-driven sunquake events in Table 1, none of which shows a spatial correspondence with an MI region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Whether it is of physical significance or just a coincidence, we need more observations to address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' This study only covers a sample of 20 events satisfying our selection criteria that the RHESSI era can provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In order to reach a more conclusive result, more events are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' RHESSI has been decommissioned since 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Fortunately, we can make use of imaging and spectroscopic observations with the Spectrometer/Telescope for Imaging X-rays (STIX) on board the newly launched Solar Orbiter (SolO) mission (Krucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2020) and the Hard X-ray Imager (HXI) on board the upcoming Advanced Space-based Solar Observatory (ASO-S) emission (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' These new observational facilities will help us better understand the origin of sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' We are grateful to the anonymous referee for his/her insightful comments and suggestions, which led to a signifi- cant improvement of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' This work was supported by National Natural Science Foundation of China under grants 11733003 and 12127901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' is also sponsored by National Key R&D Program of China under grants 2019YFA0706601 and 2020YFC2201201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' SDO is a mission of NASA’s Living With a Star (LWS) program.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2015, ApJ, 806, 171, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='1088/0004-637X/806/2/171 Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=', Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='1007/s11207-008-9216-6 Zharkova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=', & Zharkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 2007, ApJ, 664, 573, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='1086/518731 ENERGETIC ELECTRONS AS A DRIVER OF SUNQUAKES 7 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' List of the Flares under study and the Sunquake Information No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Date GOES RHESSI HXR Information HMI Sunquake Information Class Impulsive Phase Peaka E50 E300 Sunquake Correlation v35b v57b (UT) (UT) (1030 erg) (1027 erg) (Y/N) (HXR/MI) (m s−1) (m s−1) 1 2011 Feb 13 M6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='6 17:33:28–17:34:48 17:34:18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='02 N 2 2011 Feb 15 X2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='3 10:14:28–10:16:08 10:15:38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='8 Y HXR 534 344 Note— a Peak time for HXR emission at 50–100 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' b Oscillatory velocities at 3–5 MHz (v35) and 5–7 MHz (v57), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The values are adopted from CZ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' c The estimated work done by the MI-induced Lorentz force is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='8 × 1027 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 8 Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' WL and X-ray observations of the 2012 October 23 X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='8 flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' (a) time profiles of the HMI continuum intensity around 6173 ˚A (black), RHESSI HXR count rate at 50–100 keV (red), and GOES SXR flux in 1–8 ˚A (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' (b) the base- difference HMI continuum map at the continuum peak time in an inverse scale, where the dashed box encloses the main flaring region used for continuum calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Overplotted on the map is a simultaneous RHESSI 50–100 keV image reconstructed using the Pixon algorithm, with contour levels corresponding to 30%, 60%, and 90% of the maximum intensity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' For this sunquake-active flare, the location of the sunquake epicenter is also marked out with an asterisk sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' ENERGETIC ELECTRONS AS A DRIVER OF SUNQUAKES 9 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Fitting results for the RHESSI spectra taken around the HXR peak in three flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In each panel, the event number in Table 1 is labeled in the upper left, the black and grey lines in histogram mode denote the background-subtracted photon flux and the background, respectively, while the colored curves represent different components of the modeled spectrum based on the best-fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In addition, the residual between the modeled and observed spectra is plotted in the bottom part of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' 10 Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Histograms of the total energy of nonthermal electrons for the flare events with (upper) and without (lower) sunquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The left panels are for the distributions of E50 while the right for E300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Note that a bin size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content='5 dex is adopted for the histogram plotting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Locations of the MIs (orange plus yellow contours), HXR source (red contours), and sunquake epicenter (green asterisk) for two flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' In each panel the background image is a corresponding HMI continuum map, with the PIL drawn in blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' The contours for MI indicate an increase of the horizontal magnetic field at levels of 300 G and 600 G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} +page_content=' Note that the sunquake-related MI in panel (a) is highlighted in yellow contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfCgJv/content/2301.02865v1.pdf'} diff --git a/.gitattributes b/.gitattributes index 9a17be850803fea4649c4efca3c5a2ce8ef9ce1c..f204e7e1c89fbe889b9f12eacddaffdd3a53d9a7 100644 --- a/.gitattributes +++ b/.gitattributes @@ -7440,3 +7440,61 @@ StE2T4oBgHgl3EQfCAbz/content/2301.03610v1.pdf filter=lfs diff=lfs merge=lfs -tex OdFPT4oBgHgl3EQfmzXU/content/2301.13127v1.pdf filter=lfs diff=lfs merge=lfs -text sdAyT4oBgHgl3EQf0Pk5/content/2301.00713v1.pdf filter=lfs diff=lfs merge=lfs -text 9NFRT4oBgHgl3EQfqTc6/content/2301.13616v1.pdf filter=lfs diff=lfs merge=lfs -text 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sha256:6f09d63ef5d472aeab2f305e61f6ccdd93c4dbda0ca9a756bee6399bf76c0ded +size 59347 diff --git a/1NFST4oBgHgl3EQfWTh0/content/tmp_files/2301.13780v1.pdf.txt b/1NFST4oBgHgl3EQfWTh0/content/tmp_files/2301.13780v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..28dd1129f9195dd4481c79e9657a16d45b070020 --- /dev/null +++ b/1NFST4oBgHgl3EQfWTh0/content/tmp_files/2301.13780v1.pdf.txt @@ -0,0 +1,2151 @@ +Commuting Cohesions +David Jaz Myers +Mitchell Riley +February 1, 2023 +Abstract +Shulman’s spatial type theory internalizes the modalities of Lawvere’s axiomatic cohesion in a homotopy +type theory, enabling many of the constructions from Schreiber’s modal approach to differential cohomology to +be carried out synthetically. In spatial type theory, every type carries a spatial cohesion among its points and +every function is continuous with respect to this. But in mathematical practice, objects may be spatial in more +than one way at the same time; a simplicial space has both topological and simplicial structures. Moreover, +many of the constructions of Schreiber’s differential cohomology and Schreiber and Sati’s account of proper +equivariant orbifold cohomology require the interplay of multiple sorts of spatiality — differential, equivariant, +and simplicial. +In this paper, we put forward a type theory with “commuting focuses” which allows for types to carry +multiple kinds of spatial structure. The theory is a relatively painless extension of spatial type theory, and +enables us to give a synthetic account of simplicial, differential, equivariant, and other cohesions carried by the +same types. We demonstrate the theory by showing that the homotopy type of any differential stack may be +computed from a discrete simplicial set derived from the ˇCech nerve of any good cover. We also give other +examples of multiple cohesions, such as differential equivariant types and supergeometric types, laying the +groundwork for a synthetic account of Schreiber and Sati’s proper orbifold cohomology. +Contents +1 +Introduction +2 +2 +A Type Theory with Commuting Focuses +6 +3 +Specializing a Focus +11 +3.1 +Detecting Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +3.2 +Detecting Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +4 +Examples of Focuses +14 +4.1 +Real Cohesions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +4.2 +Simplicial Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +4.2.1 +The ˇCech Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +4.3 +Global Equivariant Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +4.4 +Topological Toposes +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +5 +Multiple Focuses +25 +5.1 +Commuting Cohesions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +6 +Examples with Multiple Focuses +29 +6.1 +Simplicial Real Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +6.2 +Equivariant Differential Cohesion +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +6.3 +Supergeometric Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +1 +arXiv:2301.13780v1 [math.CT] 31 Jan 2023 + +A Proof Sketches for Admissible Rules +36 +1 +Introduction +Homotopy type theory is a novel foundation of mathematics which centers the notion of identification of mathe- +matical objects. In homotopy type theory, every mathematical object is of a certain type of mathematical object; +and, if x and y are both objects of type X, then we know by virtue of the definition of the type X what it means +to identify x with y as elements of the type X. For example, if x and y were real vector spaces (so that X was the +type of real vector spaces), then to identify x with y would be to give a R-linear isomorphism between them. If x +and y were smooth manifolds, then to identify them would be to give a diffeomorphism between them. If x and +y were mere numbers, then to identify them would be simply to prove them equal. And so on, for any type of +mathematical object. +Homotopy theory, in the abstract, is the study of the identifications of mathematical objects. Homotopy type +theory is well suited for synthetic homotopy theory (e.g. [12, 24, 13, 18] and many others), but to apply these +theorems in algebraic topology — where objects are identified by giving continuous deformations of one into the +other — requires a modification to the theory. To emphasize the difference here, compare the higher inductive +circle S1, which is the type freely generated by a point with a self-identification, with the topological circle S1 +defined as the set of points in the real plane with unit distance from the origin: +S1 ≡ {(x,y) : R2 | x2 +y2 = 1}. +The base point of the higher inductive circle S1 has many non-trivial self-identifications, whereas two points of the +topological circle may be identified (in a unique way) just when they are equal. The two types are closely related +however: the higher inductive circle S1 is the homotopy type of the topological circle S1 obtained by identifying +the points of the latter by continuous deformation. Ordinary homotopy type theory does not have the language to +express this relationship, and therefore cannot apply the synthetic theorems concerning the higher inductive circle +to topological questions about the topological circle. +What is needed is a way to distinguish between types which carry topological structure and discrete types +with only homotopical structure. In his Cantor’s ‘Lauter Einsen’ and Cohesive Toposes [27], Lawvere points out +that this distinction between natively cohesive and discrete sets is already present in the writings of Cantor as the +distinction between the Menge of mathematical practice and the abstract Kardinalzahlen which arise by abstracting +away from the relationships among the points of a space. In the paper, and his subsequent Axiomatic Cohesion +[28], Lawvere formalizes this opposition between cohesion and discreteness as an adjoint triple between toposes: +Mengen +Kardinalen +points +codiscrete +discrete +This adjoint triple induces an adjoint pair of idempotent (co)monads on the topos of spaces or Mengen: the +left adjoint, ♭, retopologizes a space with the discrete topology, and the right adjoint, ♯, retopologizes it with the +codiscrete topology. Lawvere notes that in many cases — when the spaces in question are “locally connected” — +there will be a fourth adjoint π0 on the left which produces the discrete set of connected components of a space; +this system of adjoint functors characterizes his axiomatic cohesion. +But the real power of Lawvere’s axiomatic cohesion is unlocked by Schreiber’s move from 1-toposes whose +objects are cohesive sets to ∞-toposes whose objects are cohesive homotopy types. In his Differential Cohomology +in a Cohesive ∞-Topos (DCCT) [44], Schreiber shows that Lawvere’s axiomatics, when interpreted in ∞-toposes, +give rise to the hexagonal fracture diagrams which characterize differential cohomology — alongside many other +observations about the centrality of the defining adjoints of cohesion in higher topology and physics. What was +the functor π0 that took the connected components of a space becomes, in the ∞-categorical setting, the functor +2 + +Π∞ which takes the shape (in the sense of Lurie [31, §7.1.6]) of a stack. All in all, a cohesive ∞-logos has three +adjoint endofunctors +S ⊣ ♭ ⊣ ♯ +where S takes the shape or homotopy type of a higher space considered as a discrete space, ♭ takes its under- +lying homotopy type of discrete points, and ♯ takes the underlying homotopy type of points but retopologized +codiscretely. +In Brouwer’s Fixed Point Theorem in Real-Cohesive Homotopy Type Theory [47] (henceforth Real Cohesion), +Shulman brings this distinction between cohesive Mengen and discrete Karndinalen to homotopy type theory via +his spatial type theory. Spatial type theory internalizes the ♭ and ♯ modalities from Schreiber’s DCCT which relate +discrete (but homotopically interesting) types like S1 and spatial types like S1. Spatial type theory also improves +upon a previous axiomatization of these modalities in HoTT due to Schreiber and Shulman [45], by replacing +axioms with judgemental rules. Cohesive homotopy type theory is spatial type theory with an additional axiom that +implies the local contractibility of the sorts of spaces in question; from this axiom the further left adjoint S to ♭ may +be defined. +Homotopy type theory may be interpreted into any ∞-topos [26, 46], so that a type in homotopy type theory +becomes a sheaf of homotopy types externally. In particular, if we interpret the topological circle S1 defined as +a subset of R2 into the ∞-topos of sheaves on the site of continuous manifolds, it becomes the sheaf (of sets) +represented by the external continuous manifold S1, while the higher inductive circle S1 gets interpreted as the +constant sheaf at the homotopy type of the circle. By the Yoneda lemma, then, any function definable on S1 is +necessarily continuous. Since functions f : X → Y in HoTT are defined simply by specifying an element f(x) : Y +in the context of a free variable x : X, variation in a free variable confers a liminal sort of continuity: such an +expression could be interpreted in a spatial ∞-topos in which case it necessarily defines a continuous function. +Shulman’s spatial type theory works by introducing the notion of a crisp free variable to get around this liminal +continuity. An expression in spatial type theory depends on its crisp free variables discontinuously. The modalities +♭ and ♯ of spatial type theory represent crisp variables universally on the left and right respectively. In this way, ♭X +is the discrete retopologization of the spatial type X, while ♯X is its codiscrete retopologization — a map out of ♭X +is a discontinuous map out of X, while a map into ♯X is a discontinuous map into X. +Spatial type theory is intended to be interpreted into local geometric morphisms γ : E → S of ∞-toposes, +those for which γ∗ has a fully faithful right adjoint γ! which gives a geometric morphism f : S → E (with f∗ := γ!) +adjoint to γ which acts as the focal point of E as a space over S . The adjoint modalities ♭ and ♯ are interpreted +as the adjoint idempotent (co)monad pair γ∗γ∗ and γ!γ∗ respectively. A crisp free variable is then one which varies +over an object of the focal point S : a free variable is crisp when it is in focus. +There is not only one way for mathematical objects to be spatial. Spaces may cohere with smooth, analytic, +algebraic, condensed, and simplicial or cubical combinatorial structures — and more. Each of these cases would +give rise to a particular spatial type theory as the internal language of an appropriate local ∞-topos. But there are +many cases arising in practice where we need not just one axis of spatiality, but many at once. For example, it is +a classical theorem that the homotopy type of a manifold may be computed as the realization of a (topologically +discrete) simplicial set associated to the ˇCech nerve of a good open cover of the manifold. This theorem relates a +simplicial set to a continuous space, via an intermediary simplicial space which is both continuous and simplicial +at the same time — the ˇCech nerve of the cover. But in spatial homotopy type theory there is only one notion of +crisp variable, and therefore just one sort of spatiality. +For simplicial types, the discrete reflection is the 0-skeleton sk0, while the codiscrete reflection is the 0- +coskeleton. For simplicial spaces, we then have both the (topologically) discrete ♭ and codiscrete ♯, as well as +the simplicially 0-skeletal sk0 and 0-coskeletal csk0. Interestingly, the ˇCech nerve itself arises from these modal- +ities: the ˇCech nerve of a map f : X → Y between 0-skeletal types (that is, continuous or differential stacks with +no simplicial structure) is its csk0-image, as we will see later in Proposition 4.2.13. A simplicial space has both a +shape SX and a realization (or colimit) reX; the first is a topologically discrete simplicial type, while the latter is a +0-skeletal but spatial type. With all these modalities, we can prove the theorem about good covers described above +as Theorem 6.1.5. +3 + +Another use case for multiple axes of spatiality is Sati and Schreiber’s Proper orbifold cohomology [43], where +orbifolds are understood both as having both differential structure (as differential stacks) and global equivariant +structure (concerning their singularities). In order to get the correct generalized cohomology of orbifolds without +relying on ad-hoc constructions based on a global quotient presentation of the orbifold, Sati and Schreiber work +with the ∞-topos of global equivariant differential stacks, which is local both over the global equivariant topos and +the topos of differential stacks. Here the differential modalities S, ♭ and ♯ are augmented with the modalities of +equivariant cohesion [41]: +< +, +⊂ +, and +≺ +, which take the strict quotient, the underlying space as an invariant type, +and the Yoneda embedding of the underlying space of a global equivariant type respectively. Again the modalities +play a central role in the theory, with the ordinary Borel cohomology of a global quotient orbifold X �G being the +ordinary cohomology of S +⊂ +(X �G), while the proper equivariant Bredon cohomology of X �G is the cohomology +of +≺ +(X �G), twisted by the map to +≺ +BG classifying the quotient map +≺ +X → +≺ +(X �G). +In these cases, modalities that lie in the same position in their adjoint chain commute with each other, so, for +example, ♭ commutes with sk0 and ♯ commutes with csk0. However, there are cases where these modalities are +nested, with one spatiality being a refinement of another. This occurs for example in supergeometry as formulated +by Schreiber in [44] with the modalities of solid cohesion. The supergeometric focus is given by the even comodal- +ity ⇒ (which takes the even part of a superspace) and the rheonomic modality Rh which is given by localizing at +the odd line R0|1. +In this paper, we put forward a modification of spatial type theory to allow for multiple axes of spatiality. +Our theory works by allowing for a meet semi-lattice of focuses ♥,♣,..., each with a separate notion of ♥-crisp +variable and pair of adjoint (co)modalities ♭♥ and ♯♥. Like spatial type theory, our custom type theory gets us to +the coalface of synthetic homotopy theory very efficiently while staying simple enough to be used in an informal +style. +The presence of multiple notions of crispness forces a more complex context structure than spatial type theory’s +separation of the context into a crisp zone and cohesive zone. Similar to many other modal type theories [30, 22, +21, 9, 38], we annotate each variable with modal information, here, the focuses for which that variable is crisp. The +typing rules for the modalities of each focus then work essentially independently. The exception is ♭-elimination, +which is upgraded to allow the crispness of the term being eliminated to be maintained in the variable bound by +the induction (a ‘crisp’ induction principle). +Ours is far from the only extension of type theory with multiple modalities, but as we discuss in more detail +later, no existing theory has the combination of features that we are looking for: dependent types (ruling out [30]) +that may depend on modal variables (ruling out [9]), multiple commuting comodalities (ruling out [47, 11, 38]) +each with a with right-adjoint modality (ruling out [33]) and no further left-adjoints (ruling out [22, 21] and [16, +§14]). +In addition to allowing us to formalize the theorem about ˇCech nerves of open covers as Theorem 6.1.5, our +type theory will be able to handle the equivariant differential cohesion used by Sati and Schreiber in their Proper +orbifold cohomology [43], as well as the nested focuses of Schreiber’s supergeometric solid cohesion [44]. This +extends the work of Cherubini [17] and the first author [34, 35, 36] of giving synthetic accounts of the constructions +of Schreiber [44] and Sati-Schreiber [43]. +Positing an additional focus does not disturb arguments made using existing focuses, so we also expect our +theory to be helpful when dipping into simplicial arguments in the course of other reasoning by adding a simplicial +focus and making use of the new modalities. The problem of defining simplicial types in ordinary Book HoTT +remains open, and there are now a number of different approaches to constructing simplicial types which each use +some extension to the underlying type theory. In this paper, we will axiomatize the 1-simplex ∆[1] as a linear order +with distinct top and bottom and use the cohesive modalities to define the ˇCech nerve of a map and the realization +or colimit of a simplicial type. We believe our approach here would pair nicely with other approaches to simplicial +types for the purposes of synthetic (∞,1)-category theory such as [42, 14, 49, 48], where the sk0 modality would +take the core of a Rezk type.1 +1Though we have not looked in detail at how the focuses would work with the Riehl-Shulman simplicial type theory, and in particular +how they would interact with the cubes/topes zones of the Riehl-Shulman context. +4 + +Outline of the present paper. +After presenting our type theory in §2, we will look at ways to specialize the +spatiality of a focus in §3. In particular, we will observe that in many cases there is a small class of test spaces +Gi so that codiscreteness (that is, being ♯-modal) is detected by uniquely lifting against the ♭-counits ♭Gi → Gi; +such Gi will be said to detect continuity. Externally, the Gi could be any family which generates the logos under +colimits. In practice, the Gi will be test spaces which minimally carry the appropriate spatiality: in the simplicial +case, the simplices ∆[n]; in the real-cohesive case, the Euclidean spaces Rn; for condensed sets, the profinite sets, +etc. +In §3, we will also meet a family of axioms which hold for spatialities that are locally contractible. For +example, continuous manifolds which are built from Euclidean spaces by colimits are locally contractible, while +condensed sets which are built from profinite sets by colimits need not be locally contractible. In general, a space +is locally contractible when it has a constant shape in the sense of Lurie [31, §7.1.6]. +We may define a space C to be contractible when any map C → S to a discrete space S is constant. If the +converse holds — a space S is discrete (♭-modal) if every map C → S is constant — then we say that C detects the +connectivity of spaces. For example, R detects the connectivity of continuous ∞-groupoids, and ∆[1] detects the +connectivity of simplicial ∞-groupoids. If there is a space (or family of spaces) which detects connectivity, then +the local geometric morphism p corresponding to the morphism is furthermore strongly locally contractible in that +p∗ has a left adjoint p! which takes the (constant value of the) shape of a space. In the case that p is both local and +strongly locally contractible, we say that p is cohesive following Lawvere [28], Schreiber [44], and Shulman [47]. +Nullifying at the family of spaces which detect connectivity gives a modality S which is left adjoint to ♭; it may be +thought of as taking the homotopy type of a space. +In §4 we will give example axioms for specializing single focuses. We will review Shulman’s axioms for +real cohesion, where the Euclidean spaces Rn detect continuity and connectivity. We will then see simplicial +cohesion in some detail, where the simplices ∆[n] detect continuity and connectivity. We give our types simplicial +structure by axiomatizing the 1-simplex ∆[1] as a total order with distinct top and bottom elements, following +Joyal’s characterization of simplicial sets as the classifying topos for such orders [50]. We use the csk0 modality +to construct ˇCech nerves of maps. Then we will describe the global equivariant cohesion first observed by Rezk +[41] and used by Sati and Schreiber in [43]. Finally, we will briefly describe axioms for topological focuses such +as Johnstone’s topological topos of sequential spaces [25] and the condensed/pyknotic topos of Clausen-Scholze +[19] and Barwick-Haine [10]. +After surveying some of the different sorts of spatiality which types might carry, we turn our attention to +multiple focuses in §5. In Definition 5.1.3, we define what it means for two cohesions to be orthogonal: when the +family which detects the connectivity of one is discrete with respect to the other, and vice versa. We then prove a +few lemmas concerning orthogonal cohesions, in particular concerning when it is possible to commute the various +modalities past each other. +Finally, we give examples of multiple focuses in §6. We begin with simplicial real cohesion, which has both +a simplicial focus and a real-cohesive focus which are orthogonal. We prove, in Theorem 6.1.5, that the shape of +any 0-skeletal type M may be computed as the realization of a topologically discrete simplicial type constructed +from the ˇCech nerve of any good cover U of M — one for which finite intersections of the Ui are contractible in +the sense of being S-connected. +Next, we combine equivariant cohesion with differential cohesion to give the series of modalities used in Sati +and Schreiber’s Proper orbifold cohomology [43]. Happily, no extra axioms are needed to show that the two +cohesions are orthogonal; we prove this in Lemma 6.2.1. +Finally, we describe the supergeometric or “solid” cohesion of Schreiber’s Differential Cohomology in a Co- +hesive ∞-topos. This extends real cohesion with the odd line R0|1, where the “discrete” comodality of the super- +geometric focus takes the even part of a supergeometric space, and the “codiscrete” modality takes a rheonomic +reflection of the space, one whose super structure is uniquely determined by its even structure. Unlike our other +examples where the focuses involved are orthogonal, here the differential focus is included in the supergeometric +focus: any discrete space is also purely even, as is any codiscrete space. +Acknowledgements. We would like to thank Urs Schreiber for his careful reading and extensive comments during +5 + +the drafting process of the paper. And we would like to thank Hisham Sati for his feedback and words of encour- +agement. The authors are grateful for the support of Tamkeen under the NYU Abu Dhabi Research Institute grant +CG008. +2 +A Type Theory with Commuting Focuses +The fundamental duality in higher topos theory is between the ∞-topos — a general sort of space — and the ∞- +logos — the category of sheaves of homotopy types on such a space [7]. This duality is perfect: a map of ∞-toposes +E → F is defined to be a lex accessible functor Sh∞(F) → Sh∞(E ) between their corresponding ∞-logoses in the +opposite direction. +This duality between toposes and logoses gives a nice perspective on the distinction between the petite toposes, +which are used as generalized spaces in practice, and the gros toposes — or rather, their dual logoses — which +are used as categories of spaces, rather than as spaces in their own right. Quite opposite to their names, the +petite toposes are “big” spaces, while the gros toposes are “small” spaces; it is their dual logoses which are +correctly described by the adjectives “petite” and “gros”. Since the logos is the category of sheaves on the topos, +or equivalently the category of ´etale maps into the topos, the “larger” the topos the more constraining the ´etale +condition becomes. For that reason, the gros toposes have qualitatively “smaller” categories of sheaves. On the +other hand, the more general the ´etale spaces may be, the “smaller” the base topos must be. In general, the “biggest” +logoses, the logoses of spaces, must correspond to the “smallest” toposes: those toposes which are infinitesimal +patches around a focal point. This point of view is emphasized in Chapter 4 of DCCT [44]. +We may therefore, as a first pass, identify logoses of spaces as dual to those toposes E which are local over a +focal point F. A geometric morphism p : E → F is local when it admits a left adjoint right inverse f : F → E +in the (∞,2)-category of toposes which we call the focal point of p. If E is a topological space (that is, if its +corresponding logos Sh∞(E ) is the category of sheaves Sh∞(X) on a sober topological space X), then the terminal +geometric morphism γ : E → S is local just when X has a focal point: a point f ∈ X whose only open neighborhood +is the whole of X. In particular, the prime spectrum of a ring A is local if and only if A is a local ring; in this case, +the focal point is the unique maximal ideal. +On the logos side, this means that the direct image p∗ admits a fully faithful right adjoint p! (which is f∗). All +together, this gives an adjoint triple between the corresponding logoses: +Sh∞(E ) +Sh∞(F) +p∗ +p! +p∗ +Thinking of the objects of Sh∞(E ) as generalized spaces and the objects of Sh∞(F) as mere homotopy types +(sheaves on a point), we may see the direct image p∗ as taking the underlying homotopy type of points of a space, +while p∗ and p! are the discrete and codiscrete topologizations of bare homotopy types, respectively. This adjoint +triple gives rise to an adjoint pair +p∗p∗ ⊣ p!p∗ +of a idempotent comonad p∗p∗ and idempotent monad p!p∗ on the logos Sh∞(E ). Understood as operations on +spaces, these are the discrete and codiscrete retopologizations of a space respectively. +Examples of local toposes with focal point F having category of sheaves Sh∞(F) = ∞Grpd the ∞-category +of ∞-groupoids include simplicial types ∞Grpd∆op (where discrete is 0-skeletal and codiscrete is 0-coskeletal), +continuous and differentiable ∞-groupoids2 Sh∞({Rn}) (where discrete means all charts are constant, and codis- +crete means that any function valued in the set of points is a chart), condensed ∞-groupoids (where discrete means +discrete, and codiscrete means codiscrete), and global equivariant ∞-groupoids ∞GrpdGloop (where discrete means +2These are the gros toposes of C 0 and C ∞ manifolds, respectively. +6 + +being a constant presheaf on the global orbit category, and codiscrete means being a presheaf representable by an +ordinary ∞-groupoid). +Shulman [46] has shown that every ∞-logos may be presented by a model of homotopy type theory, allowing +reasoning conducted in homotopy type theory to be interpreted in any ∞-logos. In this sense, homotopy type theory +is to ∞-logoses as set theory is to the 1-logoses of Grothendieck, Lawvere, and Tierney. In Brouwer’s Fixed Point +Theorem in Real-Cohesive Homotopy Type Theory [47], Shulman also put forward a spatial type theory which +may (conjecturally) be interpreted into any local geometric morphism. Spatial type theory is characterized by +including an adjoint pair ♭ ⊣ ♯ of a lex comodality ♭ and lex modality ♯. These are to be interpreted as p∗p∗ and +p!p∗ respectively. +In spatial type theory, any type has a spatial structure. The existence of this spatial structure is witnessed by +the two opposite ways that we can get rid of it: either we can remove all the spatial relationships between points, +using the “discrete” ♭ comodality, or we can trivialize the spatial relations using the “codiscrete” ♯ modality. We +emphasize that this spatial structure is distinct from the homotopical structure that all types have by virtue of the +identifications between their elements. For example, the topological circle +S1 := {(x,y) : R2 | x2 +y2 = 1} +has a spatial structure as a subset of the Euclidean plane (as a sheaf on the site of continuous manifolds, for +example), but is a homotopy 0-type (or “set”) without any non-trivial identifications between its points; in particular +ΩS1 = ∗. The homotopy type S1 of the circle, however, is spatially discrete but has many non-trivial identifications +of its point: in particular ΩS1 = Z. +There is not, however, only one way to be spatial in mathematics. For example a simplicial topological space +has both a simplicial structure and a topological structure. This can be witnessed at the level of toposes as well. If +p : E → F admits a focal point f : F → E , then f ∆op : F ∆op → E ∆op is also a focal point of p∆op : E ∆op → F ∆op, +where the logos Sh∞(E ∆op) := (Sh∞(E ))∆op is the category of simplicial objects in the logos Sh∞(E ). But there is +another local geometric morphism γ : E ∆op → E where γ∗ sends a simplicial sheaf X• to X0 and γ! is given by the +0-coskeleton csk0 Sn := Sn. These two different axes of spatiality on the objects of Sh∞(E )∆op commute, in that the +following diagram of adjoints commutes: +Sh∞(E )∆op +Sh∞(F)∆op +Sh∞(E ) +Sh∞(F) +p∆op +∗ +γ∗ +γ∗ +p∗ +In particular, we have that p∗∆op p∗∆op and γ∗γ∗ commute as endofunctors of Sh∞(E )∆op. The former discretely +retopologizes a simplicial space, while the latter includes the space of 0-simplices as a 0-skeletal simplicial space. +Each focus gives an axis along which the objects of the top logos Sh∞(E )∆op may carry spatial structure. +When working in, say, simplicial differential spaces, we would like to have access to both the S ⊣ ♭ ⊣ ♯ of real +cohesion and the re ⊣ sk0 ⊣ csk0 of simplicial cohesion. Shulman’s spatial type theory offers no way to do this: the +♭ and sk0 comonads have incompatible claims on the notion of ‘crisp’ variable. +The solution is to allow a separate notion of crispness for each focus we are interested in. In this section, we +will describe the rules for a type theory with commuting focuses, generalizing ordinary spatial type theory in the +case of a single non-trivial focus. We will then describe axioms which make these into commuting cohesions, in +the sense of cohesive type theory. +To this end, we will fix an commutative idempotent monoid Focus of focuses; we will write the product of the +focus ♥ and the focus ♣ as ♥♣. This product induces an ordering on the focuses by saying that ♣ ≤ ♥ whenever +7 + +♣♥ = ♣. With respect to this ordering, the product becomes the meet; we may therefore also think of Focus as a +meet semi-lattice. We will write the identity element of Focus as ⊤, and note that it is the top focus with respect to +the order. +For most of our purposes in this paper, our commutative idempotent monoid Focus of focuses will be freely +generated by a finite set of basic focuses. Explicitly, we may take Focus = Pf (BasicFocuses)op to be the set of +finite subsets of the set of basic focuses with union as the product, and therefore the opposite of the ordering of +subsets by inclusion. +All variables in the context will be annotated with the focus that they are in: +x :♥ X ⊢ t : T +In general, we will abbreviate the context entry x :⊤ X as x : X. In the case that Focus = {♥ ≤ ⊤} is freely +generated by one basic focus, we recover the split context used in Shulman’s spatial type theory, where our context +x :♥ X, y :⊤ Y ctx corresponds to Shulman’s context x :: X | y : Y ctx. +To describe the typing rules, we will need a couple of auxiliary operations on contexts. The first operation ♥Γ +adds a specific focus ♥ to the annotation on every variable in a context. So: +♥(·) :≡ · +♥(Γ,x :♣ A) :≡ (♥Γ),x :♥♣ A +We also need an operation ♥ \ Γ that deletes any variables not contained within a given focus ♥; this is the +equivalent of going from ∆ | Γ ctx to ∆ | · ctx in ordinary spatial type theory. +♥\(·) :≡ · +♥\(Γ,x :♣ A) :≡ +� +(♥\Γ),x :♣ A +if ♣ ≤ ♥ +♥\Γ +otherwise +We say that a variable x :♣ X is ♥-crisp if ♣ ≤ ♥, and so the ♥-crisp variables are precisely those that survive +the ♥\Γ operation. We say that a term t : T is ♥-crisp if both it and its type T only contain ♥-crisp variables, i.e., +it is well-formed in context ♥\Γ. +We are now ready to describe the rules of the type theory. All the usual type formers — Σs, Πs, etc. — will be +included as usual, only referring to variables of the top focus ⊤. By the convention that x :⊤ X be written as x : X, +these rules look exactly as they do usually. We therefore focus on the new features of type theory with commuting +focuses. +Structural Rules. +CTX-EMPTY · ctx +CTX-EXT ♥\Γ ⊢ A type +Γ,x :♥ A ctx +VAR +Γ,x :♥ A,Γ′ ctx +Γ,x :♥ A,Γ′ ⊢ x : A +In prose, these rules read as follows: +• CTX-EMPTY: The empty context is a context. +• CTX-EXT: If A is a ♥-crisp type in context Γ, then Γ,x :♥ A ctx is a context. +• VAR: If x :♥ A appears in a context, then the variable x has type A in that context. +Remark 2.0.1. Given a context Γ,x :♥ A ctx, it must be the case that A only depends on the variables in Γ which +are themselves ♥-crisp. This careful context formation rule is what replaces the division of the context into two +zones in Shulman’s spatial type theory. In the conclusion of the variable rule, the type A is well-formed in context +Γ,x :♥ A,Γ′ ctx by the admissible DIVIDE-WK rule given below, followed by further weakening with Γ′. +8 + +Remark 2.0.2. Rather than annotating variables, may be tempting to try a floating context separator |♥ for each +focus, so that the variables to the left of |♥ are precisely the ♥-crisp ones. Such contexts are not sufficiently +general; specifically, the ♭-elimination rule will let us produce a context containing x :♥ A,y :♣ B which clearly +cannot be separated in this way. +The following rules and equations will be made admissible, with the proofs sketched in Appendix A. +WK +Γ,Γ′ ⊢J +Γ,x :♥ A,Γ′ ⊢J +−−−−−−−−−− +SUBST +♥\Γ ⊢ a : A +Γ,x :♥ A,Γ′ ⊢J +Γ,Γ′[a/x] ⊢J [a/x] +−−−−−−−−−−−−−−−−−−−−− +PROMOTE-CTX +Γ ctx +♥Γ ctx +−−−− +PROMOTE +Γ ⊢J +♥Γ ⊢J +−−−−− +DIVIDE-CTX +Γ ctx +♥\Γ ctx +−−−−−− +DIVIDE-WK +♥\Γ ⊢J +Γ ⊢J +−−−−−− +♥(♣Γ) ≡ (♥♣)Γ +♥\(♣\Γ) ≡ (♣♥)\Γ +• First, we have ordinary weakening by a variable, and a ‘crisp’ substitution similar to that used in spatial +type theory, where crisp variables may only be substituted with similarly crisp terms. These specialize to the +ordinary weakening and substitution rules when used for ♥ ≡ ⊤. +• PROMOTE-CTX corresponds to the application of the endofunctor ♥ to the context Γ, and PROMOTE to +precomposition with the counit morphism ♥Γ → Γ. +• DIVIDE-CTX gives the largest ‘subcontext’ ♥ \ Γ of Γ such that there is a substitution Γ → ♥(♥ \ Γ). The +context operation ♥ \ − thus acts like a left-adjoint to ♥−, although semantically a left-adjoint may not +exist. +Rules for ♭. +We now come to the rules for the ♭ comodality. +♭-FORM ♥\Γ ⊢ A type +Γ ⊢ ♭♥A type +♭-INTRO ♥\Γ ⊢ M : A +Γ ⊢ M♭♥ : ♭♥A +♭-ELIM +♣♥\Γ ⊢ A type +Γ,x :♣ ♭♥A ⊢ C type +♣\Γ ⊢ M : ♭♥A +Γ,u :♣♥ A ⊢ N : C[u♭♥/x] +Γ ⊢ (let u♭♥ := M inN) : C[M/x] +♭-BETA +♣♥\Γ ⊢ A type +Γ,x :♣ ♭♥A ⊢ C type +♣♥\Γ ⊢ K : A +Γ,u :♣♥ A ⊢ N : C[u♭♥/x] +Γ ⊢ (let u♭♥ := K♭♥ inN) ≡ N[K/u] : C[K♭♥/x] +In prose, these rules read as follows: +• ♭-FORM: If A is a ♥-crisp type, then we may form ♭♥A type. +• ♭-INTRO: If M is a ♥-crisp term of type A, then we may form M♭♥ of type ♭♥A. +• ♭-ELIM: If C is a type depending on the ♣-crisp variable x :♣ ♭♥A, and M : ♭♥A is a ♣-crisp element of +type ♭♥A, then we may assume that M is of the form u♭♥ for a ♣♥-crisp variable u :♣♥ A when defining an +9 + +element of C[M/x]. We write this element as (let u♭♥ := M inN) : C[M/x] where N : C[u♭♥/x] is the element +we defined assuming that M was of the form u♭♥. The equation ♥\(♣\Γ) ≡ (♣♥)\Γ is necessary here to +know that the type ♭♥A is well-formed in context ♣\Γ. +• ♭-BETA: If M actually is of the form K♭♥ for suitably crisp K, then we simply substitute K in for u. The term +K must be ♣♥-crisp for both the ♭-INTRO and ♭-ELIM to have been applied, and so its substitution for the +♣♥-crisp variable u is well-formed. +Remark 2.0.3. These rules are stronger than the ones used by Shulman for spatial type theory, even in the case of +a single focus. We have built in a ♣-crisp induction principle for ♭♥, for any two focuses ♥ and ♣: if the term we +are inducting on is already ♣-crisp, then we may maintain that crispness in the new assumption u. +If we have a single non-trivial focus ♥, as is the case in Shulman’s type theory, then taking ♣ = ♥ in the above +expression yields the ‘crisp ♭ induction’ principle of [47, Lemma 5.1]. This induction principle is proven by taking +a detour through ♯, but here we choose to build it into the rule directly. +Our elimination rule is in fact also admissible from the less general one that requires the freshly bound variable +to only be ♥-crisp, but we choose the more general rule for convenience. +Rules for ♯. +The rules for ♯ are a little simpler, and in the case of a single focus specialize exactly to the rules of +spatial type theory. +♯-FORM ♥Γ ⊢ A type +Γ ⊢ ♯♥A type +♯-INTRO +♥Γ ⊢ M : A +Γ ⊢ M♯♥ : ♯♥A +♯-ELIM ♥\Γ ⊢ N : ♯♥A +Γ ⊢ N♯♥ : A +♯-BETA +♥\Γ ⊢ M : A +Γ ⊢ (M♯♥)♯♥ ≡ M : A +♯-ETA +Γ ⊢ N : ♯♥A +Γ ⊢ N ≡ (N♯♥)♯♥ : ♯♥A +In prose, these rules read as follows: +• ♯-FORM: When forming the type ♯♥A, all variables may be used in A as though they are ♥-crisp. +• ♯-INTRO: When forming a term M♯♥ : ♯♥A, all variables may be used in M as though they are ♥-crisp. +• ♯-ELIM: If N is a ♥-crisp element of ♯♥A, we may extract an element N♯♥ : A. +• ♯-BETA: If M is a ♥-crisp element of A, then M♯♥♯♥ ≡ M. +• ♯-ETA: Any term of N : ♯♥A is definitionally equal to N♯♥ +♯♥. As in ordinary spatial type theory, the term N♯♥ +may not be well-typed on its own, because it may use non-crisp variables of the context Γ. It is however +well-typed underneath the outer (−)♯♥, since the introduction rule allows us to use any variable as though it +is ♥-crisp. +Remark 2.0.4. Perhaps surprisingly, the shape of the ♯-FORM and ♯-INTRO rules is what builds the left-exactness +of ♭ into the theory. This is the case even in ordinary spatial type theory, not a feature that only appears in this +multi-focus setting. The trick is that the promotion operation ♥Γ distributes over the context extensions in Γ rather +than being a ‘stuck’ context former applied to Γ as a whole. Specifically, when using ♯ to derive crisp Id-induction, +one applies ♯ to a type +x :: A,y :: A, p :: (x = y) | · ⊢ C type, +yielding a type +· | x : A,y : A, p : (x = y) ⊢ ♯C type. +Internalized, the former context represents the type (x : ♭A)×(y : ♭A)×♭(x♭ = y♭), but ♯-FORM treats it as identical +to ♭((x : A)×(y : A)×(x = y)) when applying adjointness. +10 + +Remark 2.0.5. In most cases of interest, our commutative idempotent monoid of focuses is freely generated by +a finite set of basic focuses. In this situation, it suffices to provide the ♭ and ♯ only for the basic focuses, as the +remainder can be derived. The top focus ⊤ (which semantically corresponds to the entire topos we are working in) +has both ♭⊤A and ♯⊤A canonically equivalent to A. And given focuses ♥ and ♣, it is quickly proven that ♭♥♣ is +equivalent to ♭♥♭♣ and similarly for the ♯s. +Related Type Theories. +Besides the original spatial type theory, there are several other dependent modal type +theories that come close to our needs. +The ‘adjoint type theory’ perspective [40, 29, 30] was the guiding principle that led to the original spatial type +theory of [47]. Indeed, when instantiated with appropriate mode theory, the framework of [30] reproduces a simply +typed version of the theory presented here. The specific mode theory to be used is a cartesian monoid with a system +of commuting, product-preserving endomorphisms. A dependently typed variant of adjoint type theory is not yet +forthcoming, but we expect that our dependent type theory would be an instance of it. +An separate line of work on modal type theories is Multimodal Type Theory [22, 23]. In MTT, every mode +morphism µ is reified in the type theory as a positive type former, and each modality modµ must have a left- +adjoint-like context operator written �µ. If we do not assume the existence of S, then we are only able to describe +♯ in this way. +Later work [21] describes a multimodal type theory where each mode morphism becomes a (more convenient) +negative type former. The semantic requirements are even stronger: the functor corresponding to the modality +must be a dependent right-adjoint [11], whose left adjoint is itself a parametric right adjoint. This is too strict even +to capture ♯ without additional assumptions. +In [16, §14], an alternative ‘cohesive type theory’ is presented, using a combination of the above two styles +of modal operator. Rather than working with the endofunctors on the topos of interest, the cohesive setting is +kept as an adjoint quadruple Π0 ⊣ Disc ⊣ Γ ⊣ CoDisc. A positive type former is used for Disc and negative type +formers for Γ and CoDisc, due to the requirements on having one or two left-adjoints. It is likely that this could +be extended to commuting cohesions, but the interactions of the various context �− operations for the left-adjoints +may be difficult to describe. +The type theory with context structure most formally similar to ours is ParamDTT [38, 37], where variables +in the context annotated with a modality indicate a variable under that modality directly, not its left adjoint. It +is from this work that we take the left-division notation − \ Γ for the clearing operation on contexts, which itself +has appeared in other guises, for example [39, 2, 3]. ParamDTT uses a fixed ‘mode theory’ with three modalities +{¶,id,♯} equipped with a particular composition law, but it is clear that the rules for contexts and basic type formers +would work equally well for other sets of modalities. A version of the cohesive ♭ can be derived from the ‘modal +Σ-type’, fixing the second component to be the unit type. There does not appear to be a way to derive the ordinary +(negative) rules for ♯ in ParamDTT. +3 +Specializing a Focus +A focus gives a specific axis along which a type may be spatial. In simplicial cohesion, we have a simplicial focus +sk0 ⊣ csk0 and in differential cohesion a differential focus ♭ ⊣ ♯. But what makes the simplicial focus simplicial and +the differential focus differential? In this section, we will investigate two axioms schemes which can determine the +peculiarities of a given focus. In the next section, we will see these axioms in use. +First, we note that with a single focus, type theory with commuting focuses is the same theory as Shulman’s +spatial type theory in [47]. +Theorem 3.0.1. Any of the lemmas and theorems proven in §3, 4, 5, and 6 of Real Cohesion [47] concerning ♭ +and ♯ and using no axioms are true also of ♭♥ and ♯♥ for any fixed focus ♥. Theorems which do involve the use of +axioms are also valid, so long as the crispness used in those axioms is interpreted as ♥-crispness. +Proof. The rules for ♭♥ and ♯♥ specialize to Shulman’s rules, and therefore his proofs carry through directly. +11 + +Specifically, ♭♥ is a coreflector and ♯♥ is a monadic modality, both are lex, and ♭♥ is (♥-crisply) left-adjoint to +♯♥. +Since adding a focus only expands the rules of the type theory and does not restrict the application of any of +the rules for any of the other focuses, any of the theorems proven in this section for a single focus will apply when +working with multiple focuses as well. +For the rest of this section, we will work within a single focus ♥, and for that reason we will drop the anno- +tations by ♥ in our expressions. For example, we will write ♭♥ as simply ♭, and we will write x :♥ X as x :: X, +following Shulman. +3.1 +Detecting Continuity +In this section, we will look at an axiom which ties the liminal sort of “continuity” implied by the crisp variables +of the type theory to the concrete continuity of a particular type G. +Our axiom will take the form of a lifting property characterizing those crisp maps which are ♯-modal. As we +will show in the upcoming Theorem 3.1.2, a crisp map is ♯-modal if and only if it lifts crisply (in a sense made +precise in Definition 3.1.1) against all of the ♭-counits. +Definition 3.1.1. Let c :: A → B and f :: X → Y be crisp maps. We say that c lifts crisply against f if for any crisp +square as on the left below, there is a unique crisp filler. +A +X +B +Y +c +f +∀ +∀ +∃! +♭(XB) +♭(XA) +♭(Y B) +♭(Y A) +♭(◦ f) +♭(c◦) +♭(◦f) +♭(c◦) +⌟ +More formally, we write c ⊥♭ f for the proposition that the square on the right is a pullback. +Theorem 3.1.2. A crisp map f :: X → Y is ♯-modal if and only if for all crisp A, (ε : ♭A → A) ⊥♭ f. +Proof. If f is ♯-modal, then since ♯ is lex, it lifts on the right against all ♯-equivalences. For any crisp A, the ♭-counit +ε : ♭A → A is a ♯-equivalence by [47, Theorem 6.22]. Therefore, the square +XA +X♭A +Y A +Y ♭A +f◦ +◦ε +f◦ +◦ε +⌟ +is a pullback, and since ♭ preserves crisp pullbacks ([47, Theorem 6.10]), we see that ε ⊥♭ f. +On the other hand, suppose that f lifts crisply on the right against all ♭-counits. To show that f is ♯-modal, it +will suffice to show that its ♯-naturality square is a pullback. Let X → ♯X ×♯Y Y be the gap map of the ♯-naturality +square of f, seeking to show that this map is an equivalence. It suffices to split the gap map over the naturality +square, by the universal property of the pullback. So, consider the crisp square +♭(♯X ×♯Y Y) +X +♯X ×♯Y Y +Y +snd +ε +f +F +k +where F(t) :≡ (let t := p♭ in(fst p)♯) is a version of the first projection. To check that the square commutes, it +suffices by ♭-induction to give, for crisp elements u :: ♯X, y :: Y, and p :: (♯f(u) = y♯), a term of type f(u♯) = y. But +we have crisply that ♯f(u) ≡ ♯f(u♯♯) = (f(u♯))♯ by the definition of ♯f, and composing this path with p we know +12 + +p′ :: (f(u♯))♯ = y♯. By the lexness of ♯, we therefore also have p′′ :: ♯(f(u♯) = y), so that the square commutes by +p′′♯. +By hypothesis, there is a unique crisp map k : ♯X ×♯Y Y → X filling this square. The bottom triangle says +precisely that k lives over the second projection. We will turn the top triangle into a proof that k lives over the first +projection. +Let (u,y, p) : ♯X ×♯Y Y, seeking to show that k(u,y, p)♯ = u. This latter type of paths is codiscrete (because +♯X is codiscrete), and so when mapping into it we may assume by that u is of the form x♯, reducing our goal to +k(x♯,y, p)♯ = x♯. By the lexness of ♯, it suffices to give an element of ♯(k(x♯,y, p) = x), and for this it suffices to give +an element of k(x♯,y, p) = x under the hypotheses that x, y, and p are crisp. In this case, (x♯,y, p)♭ : ♭(♯X ×♯Y Y), +and so we have that k(x♯,y, p) = F((x♯,y, p)♭) by the upper triangle. But by definition, F((x♯,y, p)♭) ≡ x♯♯ ≡ x, so +that we have succeeded in giving the desired identification. +We have shown that k lives over the naturality square; now we need to show that it splits the gap map X → +♯X ×♯Y Y. To that end, consider the following diagram: +♭X +♭(♯X ×♯Y Y) +X +X +♯X ×♯Y Y +Y +gap +♭gap +ε +ε +f +f +k +Showing that the diagram commutes as drawn follows easily by ♭-induction. We then have two crisp fillers of the +outer square: first we have idX : X → X and k ◦gap : X → X. By the uniqueness of crisp fillers, we conclude that +these must be identical. +Knowing that the crisp ♯-modal maps may be characterized by lifting crisply against ♭-counits suggests that we +could axiomatize the particular qualities of ♯ by restricting the class of ♭-counits which it suffices to lift against. To +that end, we make the following definition. +Definition 3.1.3. Let G :: I → Type be a crisp type family indexed by a ♭-modal and inhabited type I. We say that +G detects continuity when, for every crisp map f :: X → Y, +{ f is ♯-modal} +�(ε : ♭Gi → Gi) ⊥♭ f +for all i :: I +� +Remark 3.1.4. Thinking externally, it is straightforward to see that any family Gi which generates a local topos +E in question under colimits will detect continuity for the focus given by the terminal map of toposes. This is +because ♭, as a left adjoint, commutes with all colimits; therefore the problem of lifting against the ♭-counit of any +object of E can be reduced to that of lifting against the ♭-counits of the generators Gi. +Lemma 3.1.5. A crisp type X is ♯-modal if and only if ♭(A → X) → ♭(♭A → X) is an equivalence for all crisp types +A, and if G detects continuity then it suffices to check for each Gi. +Proof. When f : X → 1, the square defining crisp lifting is a pullback iff the top map is an equivalence. +If a family detects continuity, then it is a separating family for crisp maps in the following precise sense. +Theorem 3.1.6. Suppose that G :: I → Type detects continuity. Let f :: X → Y be a crisp map for which ♭f : ♭X → +♭Y is an equivalence and for all i :: I, the induced map ♭(Gi → X) → ♭(Gi → Y) given by post-composing with f is +an equivalence. Then f is an equivalence. +13 + +Proof. First, note that f is a ♯-equivalence since it is by hypothesis a ♭-equivalence. It therefore suffices to show +that f is ♯-modal, which by the assumption that G detects continuity means showing that f lifts crisply against all +♭-counits ♭Gi → Gi for i :: I. Consider the following diagram: +♭(XGi) +♭(X♭Gi) +♭(♯XGi) +♭(Y Gi) +♭(Y ♭Gi) +♭(♯Y Gi) +♭(f◦) +♭(f◦) +∼ +∼ +♭(♯ f◦) +The square on the left is the one we need to show is a pullback. For this, it will suffice to show that the middle +map ♭(f◦) : ♭(X♭Gi) → ♭(Y ♭Gi) is an equivalence, since the leftmost vertical map is an equivalence by hypothesis +and any square with two sides equivalences is a pullback. +For that, the middle vertical map is equivalent by the adjunction ♭ ⊣ ♯ to the rightmost vertical map ([47, +Corollary 6.26]). But the rightmost vertical map is an equivalence because it is post-composition by the equivalence +♯f. +3.2 +Detecting Connectivity +A focus is said to be cohesive if ♭ has a further left adjoint S which is itself a modality: +♭(SA → X) ≃ ♭(A → ♭X). +This adjunction only determines S for crisp types. It is better to define S by nullifying a family of objects; then +S is determined for all types (of any size). To this end, we make the following definition. +Definition 3.2.1. Let G :: I → Type be a crisp type family indexed by a ♭-modal type. We say that G detects +connectivity when, for any crisp type X, +{X is ♭-modal} +�X is Gi-null +for all i :: I +� +If G detects connectivity, then S is defined to be nullification at the family Gi. +Remark 3.2.2. In Real Cohesion [47], the assertion that a given family G detects connectivity is known as Axiom +C0. In [44, Definition 5.2.48], a single object with this property is said to ‘exhibit the cohesion’. +If there is a family G which detects connectivity, then we say that the focus is cohesive. This is justified by the +following theorem, which we may import directly from Real Cohesion [47]. +Theorem 3.2.3. Suppose that G detects connectivity. Then a crisp type is S-modal if and only if it is ♭-modal, and +furthermore S is crisply left-adjoint to ♭: +♭(SA → X) → ♭(A → ♭X) +Proof. This is [47, Theorem 9.15]. +4 +Examples of Focuses +To keep the various operators visually distinct, we will use completely different symbols for each focus we are +interested in. The rules governing the type formers are unchanged. +• S ⊣ ♭ ⊣ ♯ denotes real cohesion, where a set of real numbers (possible the Dedekind reals or an axiomatically +asserted set of “smooth reals”) detects connectivity. +14 + +• re ⊣ sk0 ⊣ csk0 denotes simplicial cohesion, where the (axiomatically asserted) 1-simplex ∆[1] detects con- +nectivity. +• +< +⊣ +⊂ +⊣ +≺ +denotes global equivariant cohesion, where connectivity is detected by +≺ +BG for finite groups +G. This notational convention follows Sati and Schreiber [43]. +• Various topological toposes exhibit spatial type theory, with ♭ ⊣ ♯ retopologizing types with the discrete and +codiscrete topologies respectively. In particular, Johnstone’s topological topos has a focus whose continuity +is detected by the walking convergent sequence N∞, which may be constructed as the set of monotone +functions N → {0,1}. +4.1 +Real Cohesions +In Real Cohesion [47], Shulman gives the axiom R♭ which states that a crisp type is ♭-modal if and only if it is +RD-null, where RD is the set of Dedekind cuts. In the terminology of Definition 3.2.1, this says that RD detects +continuous real connectivity. +Axiom 1 (Continuous Real Cohesion). We assume that RD detects continuous real connectivity, and also Shul- +man’s Axiom T: For every x : RD, the proposition (x > 0) is ♯-modal. +Remark 4.1.1. Though Shulman does not consider this axiom, we may also add the assumption that the family +Rn +D detects continuous real continuity. Using this assumption, we may internalize the arguments of Example 8.33 +of [47] to show that the mysterious Axiom T follows from the proposition that if f :: Rn +D → RD is crisp and +f(x) > 0 for any crisp x :: Rn +D, then in fact f(x) > 0 for all (not necessarily crisp) x : Rn +D. Since, by Corollary +8.28 of [47] (assuming the crisp LEM or the axiom of countable choice), any crisp Dedekind real is a Cauchy real, +we are equivalently asking if a function f : RD → RD is positive on all Cauchy reals, is it always positive. This +seems obvious, but as Shulman notes in Example 8.34, this obvious statement is not always true; though assuming +countable choice it is likely provable. +There are continuous but non-differentiable functions f : RD → RD. If we want to work in a topos where the +types have a smooth structure instead of just a continuous structure, then we must work with a type of smooth reals +RS. The most common way to axiomatize the type of smooth reals is using the Kock-Lawvere axiom and the other +axioms of synthetic differential geometry. See, e.g. Section 4.1 of [36] for a list of these axioms. In any case, if +RS is a type of smooth reals, then we will take differential cohesion to mean that RS detects connectivity. +Axiom 2 (Differential Real Cohesion). If RS is a type of smooth reals (say, from synthetic differential geometry), +then we assume that RS detects differential connectivity. +4.2 +Simplicial Cohesion +There is a well known difficulty in describing simplicial types in ordinary homotopy type theory — the infinite +amount of coherence data is difficult to describe formally given the tools of type theory. This difficulty has led +to extensions of type theory such as two-level type theories [5, 8, 4] which augment HoTT with strict equalities +which can be use to define simplicial homotopy types satisfying the simplicial identities strictly, bypassing the +problematic tower of coherences. +Another approach to avoiding simplicial difficulties is to simply interpret type theory into a topos of simplicial +homotopy types, rather than mere homotopy types. This is the approach taken by Riehl and Shulman in [42], where +they present a type theory that makes every type into a simplicial type and has as primitives the simplices ∆[n], so +that a simplex in a type A is a function σ : ∆[n] → A. +In this section we will also work with simplicial homotopy types, but by different means to Riehl and Shulman’s +type theory. Instead, we will describe simplicial cohesion, with adjoint modalities re ⊣ sk0 ⊣ csk0. These are +defined semantically as follows: +15 + +• The (“simplicial flat”) 0-skeletal comodality X �→ sk0 X sends a simplicial type to its 0-skeleton: +... +X2 +X1 +X0 +sk0 +�−−−−−→ +... +X0 +X0 +X0 +• The (“simplicial sharp”) 0-coskeletal modality X �→ csk0 X sends a simplicial type to its 0-coskeleton: +... +X2 +X1 +X0 +csk0 +�−−−−−−→ +... +X0 ×X0 ×X0 +X0 ×X0 +X0 +• The (“simplicial shape”) realization modality X �→ reX sends a simplicial type to its realization (or homotopy +colimit), considered as a 0-skeletal simplicial type: +... +X2 +X1 +X0 +re· +�−−−−−→ +... +colimXn +colimXn +colimXn +Because simplicial sets are the classifying (0-)topos for strict intervals (totally ordered sets with distinct top +and bottom elements) [50], and since the ∞-topos of simplicial homotopy types is the enveloping ∞-topos3 of +simplicial sets [6], we may assume the existence of an interval ∆[1] to make sure that our type theory is interpreted +in an ∞-topos equipped with a geometric morphism to S ∆op. We may then define the n-simplex ∆[n] to be the +n-length increasing sequences in ∆[1], and define an n-simplex in a type X to be a map ∆[n] → X. +Axiom 3 (Simplicial Axioms). We presume that ∆[1] is a total order with distinct top and bottom elements which +we call 1 and 0 respectively. Explicitly, this means that we have elements 0, 1 : ∆[1] and a proposition x ≤ y : Prop +for x, y : ∆[1]. This order must satisfy the following axioms: +1. For all x, x ≤ x. +2. For all x, y, and z, if x ≤ y and y ≤ z then x ≤ z. +3. For all x and y, if x ≤ y and y ≤ x, then x = y. +4. For all x, y, either x ≤ y or y ≤ x. +5. For all x, 0 ≤ x and x ≤ 1. +6. 0 ̸= 1. +3The enveloping ∞-topos of a topos is its free (homotopy) cocompletion, fixing existing homotopy colimits. +16 + +From these axioms, we may define the other simplices ∆[n] to be the chains of length n in ∆[1]: +∆[n] :≡ {⃗x : ∆[1]n | x1 ≤ x2 ≤ ··· ≤ xn}. +We also assume the following: +• (Axiom ∆sk0) ∆[1] detects simplicial connectivity: a simplicially crisp type X is 0-skeletal if and only if +every map ∆[1] → X is constant. +• The family ∆[−] : N → Type detects simplicial continuity. +• (Axiom ∂∆) For i : ∆[1], we have csk0((i = 0)∨(i = 1)). +• Each ∆[n] is crisply projective. That is, for a simplicially crisp E : ∆[n] → Type, we have a map +sk0((i : ∆[n]) → ∃Ei) → ∃sk0((i : ∆[n]) → Ei). +As there is an obvious map the other way, this map is an equivalence. +Let us quickly set the stage by proving that the n-simplices have trivial geometric realization and the 0-skeleton +of the n-simplex ∆[n] is the ordinal [n] ≡ {0,...,n}. +Lemma 4.2.1. The order ∆[1] has finite meets and joins, and they distribute over each other. Moreover, the +inclusion {0,1} �→ ∆[1] is an inclusion of lattices. +Proof. Suppose that x,y : ∆[1]. Then either x ≤ y or y ≤ x. In the former case, define x∧y :≡ x and x∨y :≡ y, and +in the latter case x∧y :≡ y and x∨y :≡ x. If both hold, then x = y and the definitions agree. +If x,y,z : ∆[1], then these three may find themselves in any of 6 orderings. One may then check that in each of +these cases, meets distribute over joins and vice versa. For example, supposing that x ≤ y ≤ z, then x ∧ (y ∨ z) = +x∧z = x, while (x∧y)∨(x∧z) = x∨x = x. +Lemma 4.2.2. The n-simplex ∆[n] is a retract of the n-cube ∆[1]n. Moreover, the inclusion [n] �→ ∆[n] given by +i �→ 0 ≤ ··· ≤ 0 ≤ +i times +� +�� +� +1 ≤ ··· ≤ 1 +is a retract of the inclusion {0,1}n → ∆[1]n. +Proof. Given x1,...,xn : ∆[1], define m1 :≡ � +i:n xi and let i1 : n be its index, then m2 :≡ � +n\{m1} xi, and so on. Note +that m1 ≤ m2 ≤ ··· ≤ mn, so that m : ∆[n]. Finally, if the xi were already in increasing order, then mi = xi, showing +that this is a retract. +We note that this retract argument works just as well on {0,1}n → [n], if we identify [n] with the subset +{⃗x : {0,1}n | x1 ≤ ··· ≤ xn} of increasing sequences. Since it only makes use of the lattice structure of {0,1}n and +∆[1]n, and the inclusion is a lattice homomorphism, we conclude that the necessary squares commute. +Theorem 4.2.3. The n-simplex has trivial realization: re∆[n] = ∗. +Proof. The realization re∆[n] is a retract of the realization re∆[1]n, and this is contractible since ∆[1] detects +simplicial connectivity. +Theorem 4.2.4. The inclusion [n] �→ ∆[n] given by +i �→ 0 ≤ ··· ≤ 0 ≤ +i times +� +�� +� +1 ≤ ··· ≤ 1 +is a sk0-equivalence, showing that sk0 ∆[n] ≃ [n]. +17 + +Proof. We will show that [1] �→ ∆[1] is a sk0-equivalence. This will imply that [1]n �→ ∆[1]n is a sk0-equivalence; +since [n] �→ ∆[n] is a retract of this, we may conclude that it is a sk0-equivalence as well. +Since this inclusion {0,1} �→ ∆[1] is simplicially crisp, to show that it is a sk0-counit it will suffice to show that +it is a csk0-equivalence. We therefore need an inverse csk0 ∆[1] → csk0{0,1}. Since the codomain is 0-coskeletal, it +suffices to define this map on ∆[1]. So let i : ∆[1], seeking csk0{0,1}. By Axiom ∂∆, we have csk0((i = 0)∨(i = 1)), +and since our goal is 0-coskeletal, we may assume that i = 0 or i = 1. If i = 0, then we send it to 0csk0, if i = 1, +then we send it to 1csk0. +To show that this map is the inverse of csk0{0,1} → csk0 ∆[1], we may appeal to the fact that identities in a +modal type are modal, and so we may remove the csk0 around csk0((i = 0) ∨ (i = 1)) and check that the maps +invert each other on these elements, which they clearly do. +We can also define the type of n-simplices in a simplicially crisp type, and prove a few elementary lemmas +concerning the n-simplices of types. +Definition 4.2.5. Let X be a simplicially crisp type. Then define the type Xn of n-simplices in X as +Xn :≡ sk0(∆[n] → X). +If f : X → Y is a simplicially crisp map, then it induces a map fn : Xn → Yn by post-composition. +Lemma 4.2.6. Let f : X → Y be a simplicially crisp map. If fn : Xn → Yn is an equivalence for all n, then f is an +equivalence. +Proof. This is a special case of Theorem 3.1.6, noting that X0 ≃ sk0 X. +Lemma 4.2.7. Let f : X → Y be a simplicially crisp map. Then for a crisp y : ∆[n] → Y, we have +fib fn(ysk0) ≃ sk0((i : ∆[n]) → fib f (y(i))). +Proof. We compute: +fib fn(ysk0) ≡ (x : Xn)×(fnx = ysk0) +≡ (x : sk0(∆[n] → X))×let τsk0 := xin(f ◦τ)sk0 = ysk0 +≃ (x : sk0(∆[n] → X))×let τsk0 := xinsk0(f ◦τ = y) +≃ sk0((x : ∆[n] → X)×(f ◦x = y)) +≃ sk0((x : ∆[n] → X)×((i : ∆[n]) → (f(x(i)) = y(i)))) +≃ sk0((i : ∆[n]) → fib f (y(i))). +Lemma 4.2.8. Let f : X → Y be a simplicially crisp map. Then (im f)n ≃ im fn. +Proof. We use the projectivity of the simplices.4 +(im f)n ≡ sk0(∆[n] → (y : Y)×∃fib f (y)) +≃ (y : Yn)×let σsk0 := yinsk0((i : ∆[n]) → ∃fib f (σi)) +≃ (y : Yn)×let σsk0 := yin∃sk0((i : ∆[n]) → fib f (σi)) +≃ (y : Yn)×∃fib fn(y) +≡ im fn. +4This lemma is in fact equivalent to assuming the projectivity of the simplices. +18 + +The definition of the n-simplices that we gave above is simple, but it is not that straightforward to see that it is +functorial in the ordinal [n]. We can give an alternative definition of the n-simplices which makes the functoriality +evident. +Definition 4.2.9. Let Interval denote the category of intervals: totally ordered sets with distinct top and bottom. +The maps of Interval are the monotone functions preserving top and bottom. +Let FinOrd+ denote the category of finite inhabited ordinals and order preserving maps between them — the +usual “simplex category”. We denote by [n] the ordinal {0,...,n}. +We will need a standard reformulation of the category of finite ordinals in terms of intervals (see e.g. [32, +§VIII.7]) +Lemma 4.2.10. There is a contravariant, fully faithful functor ι : FinOrdop ++ → Interval sending [n] to [n+1] with +top element n+1 and bottom element 0. To a map f : [n] → [m], we define ι f : [m+1] → [n+1] by +ι f(i) :≡ +� +min{ j | i ≤ f(j)} +n+1 +if no such minimum exists. +Conversely, to a monotone map g : [m+1] → [n+1] preserving top and bottom, we define ι−1g : [n] → [m] by the +dual formula +(ι−1g)( j) :≡ max{i | g(j) ≤ i}. +We may now define the n-simplices in a way which makes clear their functoriality in the category of finite +inhabited ordinals. +Definition 4.2.11. We define the n-simplex ∆[n] to be +∆[n] :≡ Interval(ι[n],∆[1]). +Therefore, ∆ : FinOrd+ → Set gives a functor from finite inhabited ordinals to the category of sets, where ∆(f) : +∆[n] → ∆[m] is given by precomposing with ι f : [m+1] → [n+1]. +Noting that [n] ∼= Interval(ι[n],[1]) by the fully-faithfulness of ι, the inclusion of top and bottom elements [1] �→ +∆[1] induces a natural inclusion [n] �→ ∆[n] by post-composition. As we saw in Theorem 4.2.4, these inclusions are +sk0-counits. +4.2.1 +The ˇCech Complex +The 0-coskeleton modality csk0 is useful for working in simplicial cohesion since it enables us to give an easy +construction of the ˇCech complex of a map f : X → Y between 0-skeletal types. The ˇCech complex of such a map +is, externally speaking, the simplicial type formed by repeatedly pulling back f along itself: +ˇC(f) :≡ +... +X ×Y X ×Y X +X ×Y X +X +Definition 4.2.12. Let f : X → Y be a map. The ˇCech complex ˇC(f) of f is defined to be its csk0-image: +ˇC(f) :≡ (y : Y)×csk0((x : X)×(fx = y)). +19 + +We will justify this definition by calculating the type of n-simplices of ˇC(f) when both X and Y are 0-skeletal. +Proposition 4.2.13. Let f : X → Y be a simplicially crisp map between 0-skeletal types. Then +ˇC(f)n ≃ X ×Y ···×Y X ≃ (y : Y)×((x : X)×(fx = y))n+1 +is the (n+1)-fold pullback of f along itself. +Proof. We calculate: +ˇC(f)n :≡ sk0(∆[n] → ˇC(f)) +≡ sk0(∆[n] → (y : Y)×csk0((x : X)×(fx = y))) +≃ sk0((σ : ∆[n] → Y)×((i : ∆[n]) → csk0((x : X)×(fx = σi)))) +Since Y is 0-skeletal, any map ∆[n] → Y is constant, so we may continue: +≃ sk0((y : Y)×(∆[n] → csk0((x : X)×(fx = y)))) +Since, by Theorem 4.2.4, sk0 ∆[n] = [n], we may use the adjointness of sk0 and csk0 to continue: +≃ sk0((y : Y)×csk0([n] → (x : X)×(fx = y))) +Now, we may use Lemma 6.8 of [47] to pass the sk0 into the pair type, and then use that sk0 csk0 = sk0 to continue: +≃ ((u : sk0Y)×let u := ysk0 insk0([n] → (x : X)×(fx = y))) +However, all types involved are already 0-skeletal, so we may remove the sk0s: +≃ ((y : Y)×([n] → (x : X)×(fx = y))) +≃ (y : Y)×((x : X)×(fx = y))n+1 +This last type is the (n+1)-fold pullback of f along itself, displayed in terms of its diagonal map to Y. +We can prove modally that the realization of the ˇCech nerve of a map f : X → Y is the image im f of f. This +follows from Theorem 10.2 of Real Cohesion [47]. +Theorem 4.2.14. If A is 0-coskeletal, then reA is a proposition. As a corollary, re(csk0 X) ≃ ∥X∥. +Proof. In [47], this theorem is said to rely on the crisp Law of Excluded Middle. However a glance at the proof +reveals that this assumption is only used to assume the decidable equality of sk0 ∆[1]. Since we know that sk0 ∆[1] ≃ +{0,1} has decidable equality, the proof goes through without assuming crisp LEM. +Theorem 4.2.15. For a map f : X → Y, the realization re ˇC(f) of the ˇCech nerve is the realization reim f of the +image of f. If furthermore Y is 0-skeletal, then re ˇC(f) ≃ im f. +Proof. We compute: +re ˇC(f) ≡ re((y : Y)×csk0 fib f (y)) +≃ re((y : Y)×recsk0 fib f (y)) +≃ re((y : Y)× +��fib f (y) +��) +≡ reim f. +Now if Y is 0-skeletal, then by Lemma 8.17 of [47], im f is also 0-skeletal, since it is a subtype of a 0-skeletal type. +Therefore, reim f ≃ im f, so that in total re ˇC(f) ≃ im f. +20 + +As an application of ˇCech nerves, we can see how to extract coherence data for higher groups from their +deloopings. If we take the ˇCech nerve of the inclusion ptBG : ∗ → BG of the base point of the delooping of G, we +recover a simplicial type whose simplicial identities give coherences for the multiplication of G. +Proposition 4.2.16. Let G be a crisp, 0-skeletal higher group — a 0-skeletal type identified with the loops of a +pointed, 0-connected type BG. Then +ˇC(ptBG)n ≃ Gn. +Furthermore, d1 : ˇC(ptBG)2 → ˇC(ptBG)1 is the product of the projections d0 and d2 : G2 → G. +Proof. By Proposition 4.2.13, we know that +ˇC(ptBG)n ≃ (e : BG)×((x : ∗)×(ptBG∗ = e))n+1 +≃ (e : BG)×(ptBG = e)n+1 +≃ (e : BG)×(ptBG = e)×(ptBG = e)n +≃ (ptBG = ptBG)n += Gn. +In the second to last step, we contract (e : BG)×(ptBG = e). +Now, di : ˇC(ptBG)2 → ˇC(ptBG)1 is given by forgetting the ith component of the list (e,(a,b,c)) : (e : BG) × +(ptBG = e)n+1. Therefore, +d0(e,(a,b,c)) = (e,(b,c)) +d1(e,(a,b,c)) = (e,(a,c)) +d2(e,(a,b,c)) = (e,(a,b)) +Contracting away e and the first element of the pair, we get the three equations +d0(ba−1,ca−1) = cb−1 +d1(ba−1,ca−1) = ca−1 +d2(ba−1,ca−1) = ba−1 +and indeed, we have d1(ba−1,ca−1) = d0(ba−1,ca−1)d2(ba−1,ca−1). This is equivalent, but not quite the same, as +the standard presentation. It amounts to +d0(g,h) = hg−1 +d1(g,h) = h +d2(g,h) = g. +Using the ˇCech nerve, we can extract all the coherence conditions governing a homomorphism of higher +groups. We first note that the realization of the ˇCech nerve of a group is a delooping of it. +Proposition 4.2.17. Let G be a 0-skeletal higher group with simplicially crisp delooping BG, and let ˇC(G) be the +ˇCech nerve of the basepoint inclusion ptBG : ∗ → BG. Then the projection fst : ˇC(G) → BG is a re-unit. +Proof. By Theorem 5.9 of [34], BG is 0-skeletal. By Axiom ∆sk0, it is therefore re-modal. Therefore, to show +that fst : ˇC(G) → BG is a re-unit, it suffices to show that it is re-connected. Since BG is 0-connected, it suffices +to show that the fiber over the base point is re-connected, and this fiber is equivalent to csk0 G. This follows by +Theorem 4.2.14; recsk0 G is contractible. +21 + +Proposition 4.2.18. Let G and H be 0-skeletal higher groups. Then the type of homomorphisms G → H is +equivalent to the type of pointed maps ˇC(G) ·→ ˇC(H). +Proof. Recall that a homomorphism of higher groups is by definition a pointed map between their deloopings. +That is, a homomorphism ϕ : G → H is equivalently a diagram as on the left, while a pointed map between the +ˇCech nerves is a diagram as on the right: +� +� +� +� +� +� +� +� +� +� +� +∗ +∗ +BG +BH +ptBG +Bϕ +ptBH +� +� +� +� +� +� +� +� +� +� +� +?≃ +� +� +� +� +� +� +� +� +� +� +� +∗ +∗ +ˇC(G) +ˇC(H) +ptBG +f +ptBH +� +� +� +� +� +� +� +� +� +� +� +We are aiming for an equivalence between these two types, which we may present as a one-to-one correspondence. +So, to Bϕ : BG → BH and ptBG : ptBH = Bϕ(ptBG) and f : ˇC(G) → ˇC(H) and ptf : pt ˇC(H) = f(pt ˇC(G)) associate +the type +(□ : (x : ˇC(G)) → (Bϕ(fstx) = fst(fx)))×(ptBϕ ·□(pt ˇC(G)) = fst∗ ptf ) +which, diagrammatically, is the type of witnesses that the following diagram commutes: +∗ +∗ +ˇC(G) +ˇC(H) +BG +BH +f +Bϕ +fst +fst +To show that this gives a one-to-one correspondence means showing that the types of diagrams +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∗ +∗ +ˇC(G) +ˇC(H) +BG +BH +f +Bϕ +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +and +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∗ +∗ +ˇC(G) +ˇC(H) +BG +BH +f +Bϕ +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +are both contractible, the left for any homomorphism ϕ and the right for any pointed map f. +Let ϕ be a homomorphism. Since by definition ˇC(G) and ˇC(H) were the csk0-factorizations of the basepoint +inclusions, there is a unique filler of this square: +∗ +∗ +ˇC(H) +ˇC(G) +BG +BH +Bϕ +ptBG +ptBH +∃! +But this is precisely a rearrangement of the diagram on the left. +Similarly, if f is a pointed map, then re f : BG → BH makes the diagram on the right commute, and by the +universal property of the re-unit this is the unique such map. +22 + +4.3 +Global Equivariant Cohesion +In Global Homotopy Theory and Cohesion [41], Rezk shows that the ∞-topos of global equivariant homotopy types +is cohesive over the ∞-topos of homotopy types. While Rezk constructs his site out of all compact Lie groups, we +will follow Sati and Schreiber [43] in restricting our attention to the finite groups. The global orbit category Glo +is defined to be the full subcategory of homotopy types spanned by the deloopings BG of finite groups G. This +is a (2,1)-category, and the global equivariant topos is defined to be the ∞-category of homotopy type valued +presheaves on it. +There is an adjoint quadruple connecting the global equivariant topos and the topos of homotopy types: +S Gloop +S +colim +∆ +Γ +∇ +• colimX is the colimit of the functor X : Gloop → S , which takes the strict quotient of the global equivariant +homotopy type X. +• ∆S is the inclusion of constant functors: ∆X(BG) :≡ S. We will refer to such equivariant types as invariant +types. +• ΓX :≡ X(∗) is the evaluation at the point. This is known as the homotopy quotient of the global equivariant +homotopy type X. +• ∇S is the Yoneda embedding: ∇S(BG) :≡ S (BG,S). +This adjoint quadruple gives rise to the cohesive modalities +< +⊣ +⊂ +⊣ +≺ +of equivariant cohesion: +• The (“equivariant shape”) strict quotient modality X �→ +< +X sends a global equivariant type to its strict +quotient, considered as an invariant type. +• The (“equivariant flat”) homotopy quotient modality X �→ +⊂ +X sends a global equivariant type to its homotopy +quotient, considered as an invariant type. Internally speaking, we say that an equivariantly crisp type is +invariant when it is +⊂ +-modal. +• The (“equivariant sharp”) orbisingular modality X �→ +≺ +X sends a global equivariant type to its homotopy +quotient, but considered with its natural equivariance via maps from the deloopings of finite groups. +Our axioms for global equivariant cohesion are quite straightforward: +Axiom 4 (Global Equivariant Axioms). The type family +≺ +B : FinGrp → Type sending a finite group G to +≺ +BG +detects equivariant continuity and connectivity. +The types +≺ +BG for finite groups G are the orbi-singularities. By the definition above, we may recover X(BG) +(considered with its natural equivariance) as +≺ +( +≺ +BG → X). +Remark 4.3.1. The family +≺ +BG for finite groups G is a large family, but we may reduce it to a small family by +noting that the type of finite groups is essentially small. This is a useful observation, since it allows us to conclude +that +< +, defined by nullifying all +≺ +BG, is an accessible modality. +Remark 4.3.2. Global equivariant cohesion shares a feature with Shulman’s continuous real cohesion: both are +definable in the sense that the types which detect continuity and connectivity are definable without axioms in the +type theory. This is not the case for simplicial cohesion, which appears to require postulating the 1-simplex. It is +not clear to us whether there are any general features shared by definable cohesions. +23 + +In Proper Orbifold Cohomology [43], Sati and Schreiber work with equivariant differential cohesion to give +an abstract account of the differential cohomology of orbifolds. We can prove some of their lemmas easily in +global equivariant cohesion; we will return to prove the lemmas relating equivariant and differential cohesion in +the upcoming §6.2. +The following lemma appears as Proposition 3.62 in [43]. +Lemma 4.3.3. We have the following equivalences for the generic orbi-singularities +≺ +BG: +<≺ +BG ≃ ∗ +⊂≺ +BG ≃ BG +≺≺ +BG ≃ +≺ +BG. +Proof. The first equivalence follows by the assumption that +≺ +BG detects equivariant connectivity. The second +follows by combining Theorem 6.22 of [47] to see that +⊂< +BG ≃ +⊂ +BG with Theorem 5.9 of [34] to note that since +G is crisply +⊂ +-modal (as a finite set), BG is as well. The third is simply the idempotence of +≺ +. +The following lemma is a slight strengthening of Lemma 3.65 of [43]. +Lemma 4.3.4. Let X be an equivariantly crisp set. Then X is both invariant ( +⊂ +-modal) and orbi-singular ( +≺ +- +modal). +Proof. In both cases we will use that +≺ +BG detects equivariant continuity. To show that X is invariant, we must +show that the +⊂ +-counit is an equivalence. By Theorem 3.1.6, it suffices to show that the map +⊂ +( +≺ +BG → +⊂ +X) → +⊂ +( +≺ +BG → X) +is an equivalence. But +≺ +is a lex modality and BG is 0-connected; therefore, +≺ +BG is 0-connected. Furthermore, +⊂ +X is a set since X is, using Corollary 6.7 of [47]. Therefore, the above map is equivalent to +⊂ +ε : +⊂⊂ +X → +⊂ +X, +which is an equivalence. +Similarly, to show that X is orbi-singular, it suffices to show that the map +⊂ +( +≺ +BG → X) → +⊂ +(BG → X) +given by pre-composing with the +⊂ +-counit of +≺ +BG is an equivalence. But again, by the connectivity of +≺ +BG and +BG, this map is equivalent to the identity +⊂ +X → +⊂ +X, which is an equivalence. +Remark 4.3.5. Miller’s theorem (formerly the Sullivan conjecture) states that the space of maps BG → X with G +a finite group and X a finite cell complex is equivalent to X. In equivariant modal terms, this says that finite cell +complexes (the closure of the class {/0, ∗} under pushout) are +≺ +-modal. It is not likely that this theorem could be +proven on purely modal grounds. However, if Miller’s theorem were proven in ordinary HoTT, then the modal +statement could be proven in a manner similar to Lemma 4.3.4 but instead of appealing to the truncatedness of X, +appealing to the proof of Miller’s theorem (since any crisp finite cell complex is +⊂ +-modal as a crisp pushout of +⊂ +types). +4.4 +Topological Toposes +Johnstone defined his topological topos in [25] in order to provide a topos of spaces for which the geometric +realization of simplicial sets was a geometric morphism. The problem with using a real-cohesive topos for this +purpose (as suggested previously by Lawvere) is the failure of the analytic lesser limited principle of omniscience +which says that RD = (−∞,0]∪[0,∞) (see Theorem 11.7 of [47] and the discussion in Section 8.3 of ibid.). This +failure means that gluing together simplices along their (closed) faces gives the wrong topology on the resulting +space. +Johnstone remedies this by changing the test space from the real numbers to the walking convergent sequence +N∞. The walking convergent sequence may be defined internally as the set of monotone functions N → {0,1} (see +for example [20]). Therefore, it is rather straightforward to give an internal axiomatization for Johnstone’s topos. +24 + +Axiom 5 (Topological Focus). The topological focus is determined by asserting that N∞ detects topological conti- +nuity. +We may also be able to determine condensed homotopy types as in [19] — or rather the similar but more +topos-theoretic pyknotic homotopy types of [10] — using a similar axiom. Define a profinite set to be the limit of +a crisp diagram of finite sets indexed by a discrete (♭-modal) partially ordered set with decidable order. We may +then assert that the family of profinite sets detects condensed continuity. +However, we do not know if these axioms are sufficient for proving theorems in these topological toposes. +5 +Multiple Focuses +Now, we turn our attention to generalities on possible relationships between different focuses. For this section, fix +two focuses ♥ and ♣. First, we should show that the focuses do indeed commute: +Proposition 5.0.1. For any type B, the map ♯♥♯♣B → ♯♣♯♥B defined by +x �→ x♯♥♯♣ +♯♥♯♣ +is an equivalence. Furthermore, the maps ♯♥♯♣B → ♯♥♣B and ♯♣♯♥B → ♯♥♣B defined by +x �→ x♯♥♯♣ +♯♥♣ +x �→ x♯♣♯♥ +♯♥♣ +are also equivalences. +Proof. The first map is well-defined because the use of ♯♥- and ♯♣-introduction means that the assumption x +becomes crisp for both ♥ and ♣, so we may apply ♯♥- and then ♯♣-elimination to it. We may similarly define an +inverse by +x �→ x♯♣♯♥ +♯♣♯♥. +These maps are definitional inverses by the computation rules for ♯s. +The other maps are similarly well defined since being crisp for both ♥ and ♣ means being crisp for focus ♥♣. +The inverse may be defined in the straightforward way. +We also note that the ordering of focuses is reflected in the containment of their ♯-modal (and so also ♭-modal) +types. +Corollary 5.0.2. Suppose that ♣ ≤ ♥. Then any ♯♣-modal type is ♯♥-modal. +Proof. Since ♣ ≤ ♥ is defined to mean ♥♣ ≡ ♣, we know that ♯♥♣A ≡ ♯♣A. By assumption ♯♣A ≃ A, and +chaining this with the commutativity equivalence of Proposition 5.0.1 +♯♥A ≃ ♯♥♯♣A ≃ ♯♥♣A ≡ ♯♣A ≃ A. +Tracing these simple equivalences through, this does indeed give an inverse to the ♯♥-unit. +We can similarly show that the ♭s commute. This is made simpler through the use of crisp induction. +Proposition 5.0.3. Let A be an ♥♣-crisp type. Then ♭♥♭♣A → ♭♣♭♥A defined by +u �→ let v♭♥ := uin(let w♭♣ := vin(w♭♥)♭♣) +is an equivalence, natural in A. +25 + +Proof. In words, the map is defined as follows. Performing ♭♥-induction on u : ♭♥♭♣A gives an assumption v :♥ +♭♣A. A second induction on the term v : ♭♣A then gives us an assumption w :♥♣ A. This second induction is +‘♥-crisp ♭♣-induction’: the resulting assumption w inherits the ♥-crispness of term v and gains ♣-crispness from +the removal of ♭♣. Finally, we form (w♭♥)♭♣ by applying ♭-introduction twice. A map in the other direction is +constructed in the same way, and then the proofs these are inverse are immediate by another pair of inductions +each. +We note also that the ♭-inductions commute. +Lemma 5.0.4. ♭♥-induction and ♭♣-induction commute: +let u♭♣ := (let v♭♥ := M inN)inC = let v♭♥ := M in(let u♭♣ := N inC) +(when this is well-typed, i.e., v does not occur in C.) +Proof. First using uniqueness of ♭♣, we have +let u♭♣ := (let v♭♥ := M inN)inC += let w♭♥ := M in(let u♭♣ := (let v♭♥ := w♭♥ inN)inC) +≡ let w♭♥ := M in(let u♭♣ := N[w/v]inC) +≡ let v♭♥ := M in(let u♭♣ := N inC) +5.1 +Commuting Cohesions +Now let’s turn our attention to the relationships between two commuting cohesions. +Lemma 5.1.1. Suppose that ♥ and ♣ are both cohesive. If a ♥♣-crisp type A is ♭♥-modal, then S♣A is still +♭♥-modal. +Proof. We need to produce an inverse to the counit ε♥ : ♭♥S♣A → S♣A. First construct the composite +A s−→ ♭♥A +♭♥η♣ +−−−→ ♭♥S♣A +where A s−→ ♭♥A is the assumed inverse to ε♥ : ♭♥A → A. +The type ♭♥S♣A is certainly ♭♣-modal by commutativity of ♭♥ and ♭♣: +♭♣♭♥S♣A ≃ ♭♥♭♣S♣A ≃ ♭♥S♣A +and therefore the above map factors through i : S♣A → ♭♥S♣A, our purported inverse. +For one direction, consider the naturality square for the counit ε♥: +♭♥S♣A +♭♥♭♥S♣A +S♣A +♭♥S♣A +ε♥ +♭♥i +ε♥ +i +The map ♭♥i is equal to the comultiplication δ♥ : ♭♥A → ♭♥♭♥A defined by a♭♥ �→ a♭♥♭♥, because both are inverse +to the map ♭♥ε♥ : ♭♥♭♥A → ♭♥A. And so the bottom composite in the square is equal to ε♥ ◦ δ♥, which is the +identity. +26 + +In the other direction, because S♣A is S♣-modal, it suffices to show that the composite +A → S♣A → ♭♥S♣A → S♣A +is equal to the unit η♣ : A → S♣A. For this we have the following commutative diagram: +A +♭♥A +A +S♣A +♭♥S♣A +S♣A +∼ +η♣ +♭♥η♣ +ε♥ +η♣ +i +ε♥ +The left square commutes by the definition of i, the right square by naturality of ε♥, and the composite along the +top is the identity. +Lemma 5.1.2. Suppose that ♥ and ♣ are both cohesive. Then S♥S♣A → S♣S♥A is an equivalence for any ♥♣-crisp +type A. +Proof. The map η♣ ◦η♥ : A → S♥A → S♣S♥A factors through S♥S♣A, because S♣S♥A is ♭♣-modal, as a type of the +form ♭♣X, and also ♭♥-modal, by the previous lemma. The map the other way is defined similarly. To show these +are inverses it suffices to show that they become so after precomposition with the composites of the units, because +S♣S♥A and S♥S♣A are ♥♣-discrete; this is immediate by the definition of the maps. +We might also hope that, say, ♭♥ and S♣ commute in general, but there is a useful sanity check that shows this +is not possible. In the bare type theory with no axioms, there is nothing that prevents interpretation in a model +where ♭♥ ≡ ♭♣ and S♥ ≡ S♣. In ordinary cohesive type theory it is certainly not the case that ♭ and S commute, and +so ♭♥S♣ ≃ S♣♭♥ cannot be provable without further assumptions on ♥ and ♣. +A sufficient assumption on our focuses to make ♭♥ and S♣ commute in this way is the following: +Definition 5.1.3. Suppose that G :♥♣ I → Type and H :♥♣ J → Type detect ♥ and ♣ connectivity respectively. +We say that focuses ♥ and ♣ are orthogonal if Gi is ♭♣-modal for all i, and Hj is ♭♥-modal for all j. +Our present goal is to show that this indeed makes ♭♥S♣ ≃ S♣♭♥. We will in fact only use that the Gi are +♭♣-modal; of course the dual results, flipping ♥ and ♣, require the other half of orthogonality. +Lemma 5.1.4. Let ♥ and ♣ be cohesive focuses that are orthogonal. Then for any ♣-crisp A, if A is S♥-modal, +♭♣A is still S♥-modal. +Proof. Our goal is to show that ♭♣A is equivalent to Gi → ♭♣A via precomposition by Gi → 1, for any i : I. We +easily check that the type Gi → ♭♣A is ♣-discrete: for any Hj, +Hj → (Gi → ♭♣A) ≃ Hj → (Gi → ♭♣A) +≃ Gi → (Hj → ♭♣A) +≃ Gi → ♭♣A +because ♭♣A is ♣-discrete. Then, by adjointness of S♣ and ♭♣: +(Gi → ♭♣A) ≃ ♭♣(Gi → ♭♣A) +≃ ♭♣(S♣Gi → A) +≃ ♭♣(Gi → A) +≃ ♭♣A +27 + +Proposition 5.1.5 (Crisp S♥-induction). Suppose that ♥ and ♣ are cohesive and orthogonal. If B is ♥-discrete and +♣-crisp, then for any ♣-crisp A the map +♭♣(♭♣S♥A → B) → ♭♣(♭♣A → B) +given by precomposition by ♭♣η♥ : ♭♣A → ♭♣S♥A is an equivalence. +Remark 5.1.6. As a rule, crisp S-induction would be written: +♣\Γ,x :♣ S♥A ⊢ C +♣\Γ,x :♣ S♥A ⊢ w : is-♭♥-modal(C) +♣\Γ ⊢ M : S♥A +♣\Γ,u :♣ A ⊢ N : C[uS♥/x] +Γ ⊢ (let uS♥ := M inN) : C[M/x] +Proof. We deploy the usual trick for deriving crisp induction principles: using ♯♣ to move the ♭♣ out of the way. +Crucially, the previous proposition is what allows us to apply the universal property of S♥ on maps into ♯♣B. +♭♣(♭♣S♥A → B) ≃ ♭♣(S♥A → ♯♣B) +≃ ♭♣(A → ♯♣B) +≃ ♭♣(♭♣A → B) +Proposition 5.1.7. If ♥ and ♣ are orthogonal and cohesive focuses, then S♥ and ♭♣ commute on ♣-crisp types. +Proof. This is now straightforward induction on S♥ and ♭♣ in both directions, using crisp S♥-induction when +defining ♭♣S♥A → S♥♭♣A and Lemma 5.1.4 when defining S♥♭♣A → ♭♣S♥A to know that ♭♣S♥A is ♥-discrete. +Corollary 5.1.8. If ♥ and ♣ are orthogonal and cohesive focuses, then ♭♥ and ♯♣ commute on ♥♣-crisp types. +Proof. On ♥♣-crisp types, ♭♥♯♣ is right adjoint to ♭♣S♥, and ♯♣♭♥ is right adjoint to S♥♭♣. Therefore, if ♭♣S♥X ≃ +S♥♭♣ for a ♥♣-crisp type X, then also ♭♥♯♣X ≃ ♯♣♭♥. +Next we investigate a relationship between S and ♯. Again, this depends on a relationship between the families +which detect continuity and connectivity of the two focuses. +Proposition 5.1.9. Suppose that ♣ is cohesive, and that the following hold: +1. H :♥♣ I → Type detects ♣-connectivity, and I is ♭♥-modal. +2. Hi is ♭♥-modal for all i :♥ I. (In particular, if ♥ and ♣ are orthogonal) +Then if X is S♣-modal then ♯♥X is also S♣-modal. +Proof. Since H detects ♣-connectivity, it suffices to show that ♯♥S♣X is Hi-null for every i : I. Since I is ♭♥-modal, +we may assume that i :♥ I is ♥-crisp by ♭♥-induction. Then we can compute: +(Hi → ♯♥X) ≃ ♯♥(Hi → ♯♥X) +≃ ♯♥(♭♥Hi → X) +≃ ♯♥(Hi → X) +since Hi was assumed ♭♥-modal +≃ ♯♥X +since X is S♣-modal +Tracing upwards through this series of equivalences shows that the composite is indeed the inclusion of constant +functions. +28 + +On the opposite extreme of orthogonality, we can see that if the Gi which detect the connectivity of ♥ are +S♣-connected, then any S♣-modal type is S♥-modal. +Proposition 5.1.10. Suppose that ♥ and ♣ are cohesive focuses where G : I → Type detects the connectivity of +♥. Then the following are equivalent: +1. Every Gi is S♣-connected. +2. Any S♣-modal type is S♥-modal. +Proof. Suppose that Gi is S♣-connected for all i and that X is S♣-modal. We may compute: +(1 → X) ≃ (S♣Gi → X) ≃ (Gi → X) +In the first equivalence, we use that Gi is S♣-connected, and in the second that X is S♣-modal. We conclude that X +is S♥-modal. +Conversely, suppose that any S♣-modal type is S♥-modal. Then in particular S♣Gi is S♥-modal, so that the +identity map S♣Gi → S♣Gi factors through S♥S♣Gi. But by Lemma 5.1.2 we have +S♥S♣Gi ≃ S♣S♥Gi ≃ S♣∗ ≃ ∗. +Therefore, the identity of S♣Gi factors through the point, which means it is contractible. +6 +Examples with Multiple Focuses +In this section, we will see examples with multiple focuses. In particular, we will see simplicial real cohesion, +equivariant differential cohesion, and supergeometric cohesion. +6.1 +Simplicial Real Cohesion +We assume two basic focuses: the real (continuous or differential) focus S ⊣ ♭ ⊣ ♯, and the simplicial focus re ⊣ +sk0 ⊣ csk0. We will write R for whichever flavor of real numbers is used in the real cohesive focus. +We will assume both the axioms of real cohesion and simplicial cohesion, as well as the following axiom +relating the two focuses. +Axiom 6 (Simplicial Real Cohesion). We assume that the real focus and simplicial focus are orthogonal — which +is to say, R is 0-skeletal and that ∆[1] is discrete. Furthermore, we assume that S is computed pointwise: for any +simplicially crisp type X, the action (ηS)n : Xn → (SX)n of the S-unit of X on n-simplices is itself a S-unit. +Our goal in this section will be to prove that if M is a 0-skeletal type — to be thought of as a “manifold”, having +only real-cohesive structure but no simplicial structure — and U is a good cover of M — one for which the finite +intersections are S-connected whenever they are inhabited — then the homotopy type SM of M may be constructed +as the realization of a discrete simplicial set — namely, the ˇCech nerve of the open cover, with each open replaced +by the point. +Definition 6.1.1. Let M be a 0-skeletal type. A cover of M consists of a discrete 0-skeletal index set I, and a +family U : I → (M → Prop) of subobjects of M so that for every m : M there is merely an i : I with m ∈ Ui. We +may assemble a cover into a single surjective map c : � +i:IUi → M, where +� +i:I +Ui :≡ (i : I)×(m : M)×(m ∈ Ui). +A cover U : I → (M → Prop) is a good cover if for any n : N and any k : [n] → I, the S-shape of the intersection +� +i:[n] +Uk(i) :≡ (m : M)×((i : [n]) → (m ∈ Uk(i))). +is a proposition. That is, S(Uk(0) ∩···∩Uk(n)) is contractible whenever there is an element in the intersection. +29 + +We begin with a few ground-setting lemmas. +Lemma 6.1.2. Let U : I → (M → Prop) be a simplicially crisp cover, and let c : � +i:IUi → M be the associated +covering map. Consider the projection π : ˇC(c) → csk0 I defined by (m,z) �→ (fstzcsk0)csk0. Over a simplicially +crisp n-simplex k : ∆[n] → csk0 I, we have +fibπn(ksk0) ≃ +� +i:[n] +Uk(isk0). +As a corollary, we have that +ˇC(c)n ≃ (k : I[n])× +� +i:[n] +Uk(i). +Proof. We compute: +fibπn(ksk0) ≡ (x : ˇC(c)n)×(πnx = ksk0) +≃ ((m,z) : (m : M)×((i : I)×(m ∈ Ui))[n])×(πne(m,z) = ksk0) +≃ (m : M)×(K : [n] → I)×(p : (i : [n]) → (m ∈ UK(i)))×(πne(m,i �→ (K(i), p)) = ksk0) +Here, e(m,z) is image under the equivalence from Proposition 4.2.13. When all the modal dust settles, we will be +left knowing that πne(m,i �→ (K(i), p)) : sk0(∆[n] → csk0 I) is the unique correspondent to K : [n] → I under the +sk0 ⊣ csk0 adjunction. Therefore, we may contract K away with ksk0 in the above type to get: +≃ (m : M)×((i : [n]) → m ∈ Uk(isk0)). +For the next lemma, we will need to know that S commutes with csk0 on suitably crisp types. +Theorem 6.1.3. For any simplicially crisp type X, we have that Scsk0 X ≃ csk0 SX. +Proof. We know by Proposition 5.1.9 that csk0 SX is S-modal. We therefore have a map Scsk0 X → csk0 SX given +as the unique factor of csk0(−)S : csk0 X → csk0 SX. We will show that this map is an equivalence. Since it is crisp, +it suffices to show that it is an equivalence on n-simplices. To that end, we compute: +sk0(∆[n] → Scsk0 X) ≃ Ssk0(∆[n] → csk0 X) +≃ Ssk0([n] → X) +≃ sk0(S([n] → X)) +≃ sk0([n] → SX) +≃ sk0(∆[n] → csk0 SX) +It remains to show that this is indeed the right equivalence. Since the first equivalence in the series above is given +as the inverse of (−)S +n : (csk0 X)n → (Scsk0 X)n, it suffices to check that given a crisp z : ∆[n] → csk0 X, (zsk0)S +corresponds under the above equivalences to (csk0(−)S ◦z)sk0. First, we send (zsk0)S to ((z◦(−)sk0)sk0)S. Then, we +send it to ((z◦(−)sk0)S)sk0, and then to (i �→ (z(isk0)S))sk0. Finally, we map this to (i �→ ((z(isk0sk0)S))csk0 sk0, which +does equal (csk0(−)S ◦z)(i) ≡ (z(i)S)csk0 at i : ∆[n]. +Lemma 6.1.4. Let U : I → (M → Prop) be a simplicially crisp cover, and let c : � +i:IUi → M be the covering map +itself. Consider the “projection” π : ˇC(c) → csk0 I defined by (m,z) �→ (fstzcsk0)csk0. Then U is a good cover if and +only if the restriction π : ˇC(c) → imπ is a S-unit. +30 + +Proof. Since csk0 and S commute by Theorem 6.1.3 and I is discrete, csk0 I is also discrete. As the subtype +of a discrete type, imπ is discrete. Therefore, it suffices to show that π : ˇC(c) → imπ induces an equivalence +S ˇC(c) ∼−→ imπ if and only if the cover U is good. Since π is crisp, S ˇC(c) → imπ is an equivalence if and only if it is +an equivalence on all n-simplices. On n-simplices, this map (at the top of the following diagram) is equivalent to +the map on the bottom of the following diagram: +(S ˇC(c))n +(imπ)n +S(ˇC(c)n) +imπn +S +� +(k : I[n])× � +i:[n]Uk(i) +� +(k : I[n])×S +�� +i:[n]Uk(i) +� +(k : I[n])×∃� +i:[n]Uk(i) +The bottom map is an equivalence if and only if the cover is good, and so we conclude the same for the top +map. +Finally, we can piece these lemmas together for our result. +Theorem 6.1.5. Let U : I → (M → Prop) be a simplicially crisp good cover of a 0-skeletal type M. Let π : ˇC(U) → +csk0 I be the projection. Then +reimπ ≃ SM. +This exhibits the shape SM as the realization of a discrete (simplicial) set. +Proof. Since the cover is good, we have that imπ ≃ SˇC(U), so that +reimπ ≃ reSˇC(U) ≃ Sre ˇC(U) ≃ SM. +Now, since I is a set and csk0 is a lex modality, csk0 I is also a set and so imπ is a set as well. Furthermore, since +subtypes of discrete types are discrete by Lemma 8.17 of [47], and csk0 I is discrete since by Theorem 6.1.3 S and +csk0 commute on simplicially crisp types, imπ is discrete. +6.2 +Equivariant Differential Cohesion +In Proper Orbifold Cohomology, Sati and Schreiber work in equivariant differential cohesion to describe the differ- +ential cohomology of orbifolds. This cohesion involves both the equivariant focus +< +⊣ +⊂ +⊣ +≺ +and the differential +(real-cohesive) focus S ⊣ ♭ ⊣ ♯. In this section, we will assume both the axioms of equivariant cohesion and differ- +ential real cohesion. We will refer to the smooth reals by R. +Unlike the simplicial real cohesive case, we do not need to add additional axioms to ensure that the equivariant +and differential cohesion are orthogonal. +Lemma 6.2.1. Equivariant and differential cohesion are orthogonal. That is: +1. The smooth reals R are invariant ( +⊂ +-modal). +2. For any finite group G, +≺ +BG is discrete (♭-modal). +Proof. Since the smooth reals are a set, they are invariant by Lemma 4.3.4. Similarly, since BG is discrete and +hence S-modal (by Theorem 5.9 of [34]), +≺ +BG is still S-modal by Proposition 5.1.9. +31 + +The following lemma appears as Lemma 3.67 of [43], and is proven quickly with our general lemmas concern- +ing orthogonal cohesions. +Lemma 6.2.2. Suppose that X is both differentially and equivariantly crisp. Then +⊂ +SX ≃ S +⊂ +X +⊂ +♭X ≃ ♭ +⊂ +♭X +⊂ +♯X ≃ ♯ +⊂ +X. +Proof. The first equivalence follows by Proposition 5.1.7. The second equivalence follows by Proposition 5.0.3. +The third follows by Corollary 5.1.8. +6.3 +Supergeometric Cohesion +In his habilitation, Differential Cohomology in a Cohesive ∞-Topos [44], Schreiber describes an increasing tower +of adjoint modalities which appear in the setting of supergeometry. The setting for supergeometric cohesion — +called “solid cohesion” in ibid. — is sheaves on the opposite of a category of super C ∞-algebras. Schreiber calls +these sheaves super formal smooth ∞-groupoids. Specifically, the site is (the opposite of) the full subcategory of +the category of super commutative real algebras spanned by objects of the form +C ∞(Rn)⊗W ⊗ΛRq +where W is a Weil algebra — a commutative nilpotent extension of R which is finitely generated as an R-module. +The factor C ∞(Rn)⊗W is even graded, while the Grassmannian ΛRq is odd graded. See Definition 6.6.13 of [44]. +The inclusion of algebras of the form C ∞(Rn) ⊗W has a left and a right adjoint. The left adjoint is given +by projecting out the even subalgebra, and the right adjoint is given by quotienting by the ideal generated by the +odd graded elements. This gives rise to an adjoint quadruple between the resulting toposes of sheaves and thus an +adjoint triple of idempotent adjoint (co)monads on the topos of super formal smooth ∞-groupoids: ⇒ ⊣ ⇝ ⊣ Rh. +Of these, ⇒ and Rh are idempotent monads. However, ⇒ does not preserve products, and so does not give an +internal modality. The action of Rh is easy to define: +RhX(C ∞(Rn)⊗W ⊗ΛRq) := X(C ∞(Rn)⊗W). +That is, RhX is defined by evaluating at the even part of the superalgebra in the site. We may characterize it +internally by localizing at the odd line R0|1, which is the sheaf represented by the free superalgebra on one odd +generator ΛR. We turn to the internal story now. +The topos of super formal smooth ∞-groupoids also supports the differential real cohesive modalities S ⊢ ♭ ⊢ ♯. +These destroy all geometric structure — super and otherwise. For this reason, we will work with the lattice +{diff < super < ⊤} of focuses. +The modalities of the super focus are ⇝ ⊢ Rh. We will refer to +⇝ +X as the even part of X, while Rh is known as +the rheonomic modality. We assume the following axioms for supergeometric or solid cohesion. +Axiom 7 (Solid Cohesion). Solid cohesion uses the focus lattice {diff < super < ⊤}. We use the definition of real +superalgebras due to Carchedi and Roytenberg [15]. +1. We assume a commutative ring R1|0 satisfying the axioms of synthetic differential geometry (as e.g. in +Section 4.1 of [36]) known as the smooth reals or the even line. +2. We assume an R1|0-module R0|1. There is furthermore a bilinear multiplication R0|1×R0|1 → R1|0 which sat- +isfies a2 = 0 for all a : R0|1. Together these axioms imply that R1|1 := R1|0×R0|1 is a R1|0-supercommutative +superalgebra. +3. We assume the following odd form of the Kock-Lawvere axiom: For any function f : R0|1 → R0|1 with +f(0) = 0, there is a unique r : R1|0 with f(x) = rx for all x. +4. We assume that R0|1 is ⇝-connected. +32 + +5. We assume that R1|0 detects differential connectivity. +6. We assume that a type is Rh-modal if and only if it is R0|1-null. +Remark 6.3.1. It might seem prudent to instead ask that differential connectivity is detected by the family con- +sisting of both R1|0 and R0|1, since we want S to nullify all representables Rn|q, but it suffices to test with R1|0 since +R0|1 admits an explicit contraction by its R1|0-module structure (appealing to Lemma 6.10 of [34]). +By Corollary 5.0.2, any ♯-modal type is Rh-modal. But also every ♭-modal type is Rh-modal. +Lemma 6.3.2. If X is S-modal (and, in particular, if X is ♭-modal), then X is Rh-modal. +Proof. Since R0|1 is S-connected due to its explicit contraction by the scaling of its module structure, any S-modal +type is R0|1-null, and therefore Rh-modal. +References +[1] +LICS ’18. Oxford, United Kingdom: Association for Computing Machinery, 2018. ISBN: 978-1-4503-5583- +4. +[2] +Andreas Abel. “A Polymorphic Lambda-Calculus with Sized Higher-Order Types”. PhD thesis. June 2006. +URL: http://www2.tcs.ifi.lmu.de/~abel/diss.pdf. +[3] +Andreas Abel. “Polarised subtyping for sized types”. In: Mathematical Structures in Computer Science 18.5 +(2008), pp. 797–822. DOI: 10.1017/S0960129508006853. +[4] +Benedikt Ahrens et al. The Univalence Principle. 2021. arXiv: 2102.06275 [math.CT]. +[5] +Thorsten Altenkirch, Paolo Capriotti, and Nicolai Kraus. “Extending homotopy type theory with strict equal- +ity”. In: Computer Science Logic. Vol. 62. LIPIcs. Leibniz Int. Proc. Inform. Schloss Dagstuhl. Leibniz-Zent. +Inform., Wadern, 2016, Art. No. 21, 17. 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URL: http://www.numdam.org/item/CTGDC_1993__34_4_259_0/. +35 + +A +Proof Sketches for Admissible Rules +We sketch proofs that the operations on syntax are admissible, demonstrating the interesting cases; those that +involve division (CTX-EXT, ♭-FORM/INTRO/ELIM, ♯-ELIM) or promotion (♯-FORM/INTRO). +Definition A.0.1. The ♥Γ and ♥\Γ context operations extend in the obvious way to telescopes Γ′, so that +♥(Γ,Γ′) ≡ (♥Γ),(♥Γ′) +♥\(Γ,Γ′) ≡ (♥\Γ),(♥\Γ′) +Lemma A.0.2 (Weakening). Single-variable weakening is admissible. +WK +Γ,Γ′ ⊢J +Γ,w :♣ W,Γ′ ⊢J +−−−−−−−−−−− +Moreover, weakening does not change the size of the derivation tree. +Proof. Induction onJ . +Case (division). +♭-FORM ♥\(Γ,w :♣ W,Γ′) ⊢ A type +Γ,w :♣ W,Γ′ ⊢ ♭♥A type +There are two subcases: +• If ♣ ≤ ♥: then ♥\(Γ,w :♣ W,Γ′) ≡ (♥\Γ),w :♣ W,(♥\Γ′), in which case we induct on the type A +and reapply the rule. +• If ♣ ̸≤ ♥: then ♥\(Γ,w :♣ W,Γ′) ≡ (♥\Γ),(♥\Γ′) ≡ ♥\(Γ,Γ′), and so already ♥\(Γ,w :♣ W,Γ′) ⊢ +A type and we can reapply the rule. +Case (promotion). +♯-FORM ♥(Γ,w :♣ W,Γ′) ⊢ A type +Γ,w :♣ W,Γ′ ⊢ ♯♥A type +By definition ♥(Γ,w :♣ W,Γ′) ≡ ♥Γ,w :♥♣ W,♥Γ′, and so we induct on A (now weakening with a variable +of focus ♥♣). +Lemma A.0.3. For any two focuses ♥ and ♣, the context ♣\(♥Γ) is an iterated weakening of ♥(♣\Γ). +Proof. This can be checked variablewise. Given a variable x :♠ A in Γ, if it survives to ♥(♣\Γ) as x :♥♠ A, then we +must have ♠ ≤ ♣. Multiplying by ♥, it follows that ♥♠ ≤ ♥♣ ≤ ♣, and so x :♥♠ A also occurs in ♣\(♥Γ). +Lemma A.0.4. The following equations involving contexts and telescopes hold: +(♥Γ′)[s/z] ≡ ♥(Γ′[s/z]) +(♥\Γ′)[s/z] ≡ ♥\(Γ′[s/z]) +36 + +Lemma A.0.5 (Substitution). +SUBST +♣\Γ ⊢ s : S +Γ,z :♣ S,Γ′ ⊢J +Γ,Γ′[s/z] ⊢J [s/z] +−−−−−−−−−−−−−−−−−−−− +Proof. Induction onJ . +Case (variable). +Three subcases as usual, for x ∈ Γ, x ≡ z and x ∈ Γ′. The interesting one is x ≡ z: +VAR Γ,z :♣ S,Γ′ ⊢ s : S +Applying DIVIDE-WK to ♣\Γ ⊢ s : S gives Γ ⊢ s : S, which can be further weakened to Γ,Γ′ ⊢ s : S. +Case (division). +♭-FORM ♥\(Γ,z :♣ S,Γ′) ⊢ A type +Γ,s :♣ S,Γ′ ⊢ ♭♥A type +There are two subcases: +• If ♣ ≤ ♥: then ♥ \ (Γ,z :♣ S,Γ′) ≡ (♥ \ Γ),z :♣ S,(♥ \ Γ′), in which case we induct, getting (♥ \ +Γ),(♥\Γ′)[s/z] ⊢ A[s/z] type. This context is equal to ♥\(Γ,Γ′[s/z]) ctx, so we can reapply the rule. +• If ♣ ̸≤ ♥: then ♥ \ (Γ,z :♣ S,Γ′) ≡ (♥ \ Γ),(♥ \ Γ′) ≡ ♥ \ (Γ,Γ′), and so z does not occur in A, and +A ≡ A[s/z], in which case we may reapply the rule. +Case (promotion). +♯-FORM ♥(Γ,z :♣ S,Γ′) ⊢ A type +Γ,z :♣ S,Γ′ ⊢ ♯♥A type +By definition ♥(Γ,s :♣ S,Γ′) ≡ ♥Γ,s :♥♣ S,♥Γ′. Applying PROMOTE to ♣\Γ ⊢ s : S yields ♥(♣\Γ) ⊢ s : S. +By Lemma A.0.3, this can be weakened to ♣(♥\Γ) ⊢ s : S, whose context is equal to (♥♣)\(♥Γ), which +is of the correct shape to be substituted into ♥Γ,z :♥♣ S,♥Γ′ ⊢ A type. Substitution gives ♥Γ,(♥Γ′)[s/z] ⊢ +A[s/z] type, and this context is equal to ♥(Γ,Γ′[s/z]) ctx, so we can reapply the rule. +Lemma A.0.6 (Promote). +PROMOTE-CTX +Γ ctx +♣Γ ctx +−−−− +PROMOTE +Γ ⊢J +♣Γ ⊢J +−−−−− +Proof. First, PROMOTE-CTX is by induction on the length of the context. Consider a context Γ,x :♥ A ctx, so that +♥ \ Γ ⊢ A type. Applying PROMOTE to A gives ♣(♥ \ Γ) ⊢ A type, which can be weakened to (♣♥) \ (♣Γ) ⊢ +A type by Lemma A.0.3, letting us form the context ♣Γ,x :♣♥ A ctx. +Case (variable). +The variable rule is immediate, because modifying the annotation on a variable does not change whether it +is usable. +37 + +Case (division). +♭-FORM ♥\Γ ⊢ A type +Γ ⊢ ♭♥A type +Inductively ♣(♥\Γ) ⊢ A type, which can be weakened to ♥\(♣Γ) ⊢ A type, and we reapply the rule. +Case (promotion). +♯-FORM ♥Γ ⊢ A type +Γ ⊢ ♯♥A type +Inductively ♣(♥Γ) ⊢ A type, and ♣(♥Γ) ≡ ♥(♣Γ), so we may reapply the rule. +Lemma A.0.7 (Division). +DIVIDE-CTX +Γ ctx +♣\Γ ctx +−−−−−− +DIVIDE-WK +Γ ctx +♣\Γ ⊢J +Γ ⊢J +−−−−−−−−−−−− +Proof. First DIVIDE-CTX. Consider a context Γ,x :♥ A ctx, so that ♥\Γ ⊢ A type. There are two cases: +• If ♥ ≤ ♣: Then +♥\Γ ≡ (♥♣)\Γ ≡ ♥\(♣\Γ), +and so ♥\(♣\Γ) ⊢ A type is of the right shape to form ♣\Γ,x :♥ A ctx. +• If ♥ ̸≤ ♣: Then ♣\(Γ,x :♥ A) ≡ ♣\Γ is well-formed by induction. +Now DIVIDE-WK. On terms: +Case (variable). +If x :♥ A is in context ♣\Γ, then it must also be in Γ and so we may reuse the variable rule. +Case (division). +♭-FORM ♥\(♣\Γ) ⊢ A type +♣\Γ ⊢ ♭♥A type +We know ♥\(♣\Γ) ≡ ♣\(♥\Γ), and so inductively ♥\Γ ⊢ A type, and we can reapply the rule. +Case (promotion). +♯-FORM ♥(♣\Γ) ⊢ A type +♣\Γ ⊢ ♯♥A type +♥(♣ \ Γ) ⊢ A type may be weakened to ♣ \ (♥Γ) ⊢ A type (without increasing the size of the derivation). +Inductively ♥Γ ⊢ A type, and we can reapply the rule. +38 + diff --git a/1NFST4oBgHgl3EQfWTh0/content/tmp_files/load_file.txt b/1NFST4oBgHgl3EQfWTh0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..54ba6cd99d336009c52154c3be832f6071d5b932 --- /dev/null +++ b/1NFST4oBgHgl3EQfWTh0/content/tmp_files/load_file.txt @@ -0,0 +1,2251 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf,len=2250 +page_content='Commuting Cohesions David Jaz Myers Mitchell Riley February 1, 2023 Abstract Shulman’s spatial type theory internalizes the modalities of Lawvere’s axiomatic cohesion in a homotopy type theory, enabling many of the constructions from Schreiber’s modal approach to differential cohomology to be carried out synthetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In spatial type theory, every type carries a spatial cohesion among its points and every function is continuous with respect to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But in mathematical practice, objects may be spatial in more than one way at the same time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' a simplicial space has both topological and simplicial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Moreover, many of the constructions of Schreiber’s differential cohomology and Schreiber and Sati’s account of proper equivariant orbifold cohomology require the interplay of multiple sorts of spatiality — differential, equivariant, and simplicial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this paper, we put forward a type theory with “commuting focuses” which allows for types to carry multiple kinds of spatial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The theory is a relatively painless extension of spatial type theory, and enables us to give a synthetic account of simplicial, differential, equivariant, and other cohesions carried by the same types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We demonstrate the theory by showing that the homotopy type of any differential stack may be computed from a discrete simplicial set derived from the ˇCech nerve of any good cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We also give other examples of multiple cohesions, such as differential equivariant types and supergeometric types, laying the groundwork for a synthetic account of Schreiber and Sati’s proper orbifold cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Contents 1 Introduction 2 2 A Type Theory with Commuting Focuses 6 3 Specializing a Focus 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 Detecting Continuity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 Equivariant Differential Cohesion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 Supergeometric Cohesion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 32 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13780v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='CT] 31 Jan 2023 A Proof Sketches for Admissible Rules 36 1 Introduction Homotopy type theory is a novel foundation of mathematics which centers the notion of identification of mathe- matical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In homotopy type theory, every mathematical object is of a certain type of mathematical object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' and, if x and y are both objects of type X, then we know by virtue of the definition of the type X what it means to identify x with y as elements of the type X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, if x and y were real vector spaces (so that X was the type of real vector spaces), then to identify x with y would be to give a R-linear isomorphism between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If x and y were smooth manifolds, then to identify them would be to give a diffeomorphism between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If x and y were mere numbers, then to identify them would be simply to prove them equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' And so on, for any type of mathematical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Homotopy theory, in the abstract, is the study of the identifications of mathematical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Homotopy type theory is well suited for synthetic homotopy theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [12, 24, 13, 18] and many others), but to apply these theorems in algebraic topology — where objects are identified by giving continuous deformations of one into the other — requires a modification to the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To emphasize the difference here, compare the higher inductive circle S1, which is the type freely generated by a point with a self-identification, with the topological circle S1 defined as the set of points in the real plane with unit distance from the origin: S1 ≡ {(x,y) : R2 | x2 +y2 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The base point of the higher inductive circle S1 has many non-trivial self-identifications, whereas two points of the topological circle may be identified (in a unique way) just when they are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The two types are closely related however: the higher inductive circle S1 is the homotopy type of the topological circle S1 obtained by identifying the points of the latter by continuous deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Ordinary homotopy type theory does not have the language to express this relationship, and therefore cannot apply the synthetic theorems concerning the higher inductive circle to topological questions about the topological circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' What is needed is a way to distinguish between types which carry topological structure and discrete types with only homotopical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In his Cantor’s ‘Lauter Einsen’ and Cohesive Toposes [27], Lawvere points out that this distinction between natively cohesive and discrete sets is already present in the writings of Cantor as the distinction between the Menge of mathematical practice and the abstract Kardinalzahlen which arise by abstracting away from the relationships among the points of a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the paper, and his subsequent Axiomatic Cohesion [28], Lawvere formalizes this opposition between cohesion and discreteness as an adjoint triple between toposes: Mengen Kardinalen points codiscrete discrete This adjoint triple induces an adjoint pair of idempotent (co)monads on the topos of spaces or Mengen: the left adjoint, ♭, retopologizes a space with the discrete topology, and the right adjoint, ♯, retopologizes it with the codiscrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lawvere notes that in many cases — when the spaces in question are “locally connected” — there will be a fourth adjoint π0 on the left which produces the discrete set of connected components of a space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' this system of adjoint functors characterizes his axiomatic cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But the real power of Lawvere’s axiomatic cohesion is unlocked by Schreiber’s move from 1-toposes whose objects are cohesive sets to ∞-toposes whose objects are cohesive homotopy types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In his Differential Cohomology in a Cohesive ∞-Topos (DCCT) [44], Schreiber shows that Lawvere’s axiomatics, when interpreted in ∞-toposes, give rise to the hexagonal fracture diagrams which characterize differential cohomology — alongside many other observations about the centrality of the defining adjoints of cohesion in higher topology and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' What was the functor π0 that took the connected components of a space becomes, in the ∞-categorical setting, the functor 2 Π∞ which takes the shape (in the sense of Lurie [31, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6]) of a stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' All in all, a cohesive ∞-logos has three adjoint endofunctors S ⊣ ♭ ⊣ ♯ where S takes the shape or homotopy type of a higher space considered as a discrete space, ♭ takes its under- lying homotopy type of discrete points, and ♯ takes the underlying homotopy type of points but retopologized codiscretely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In Brouwer’s Fixed Point Theorem in Real-Cohesive Homotopy Type Theory [47] (henceforth Real Cohesion), Shulman brings this distinction between cohesive Mengen and discrete Karndinalen to homotopy type theory via his spatial type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Spatial type theory internalizes the ♭ and ♯ modalities from Schreiber’s DCCT which relate discrete (but homotopically interesting) types like S1 and spatial types like S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Spatial type theory also improves upon a previous axiomatization of these modalities in HoTT due to Schreiber and Shulman [45], by replacing axioms with judgemental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Cohesive homotopy type theory is spatial type theory with an additional axiom that implies the local contractibility of the sorts of spaces in question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' from this axiom the further left adjoint S to ♭ may be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Homotopy type theory may be interpreted into any ∞-topos [26, 46], so that a type in homotopy type theory becomes a sheaf of homotopy types externally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In particular, if we interpret the topological circle S1 defined as a subset of R2 into the ∞-topos of sheaves on the site of continuous manifolds, it becomes the sheaf (of sets) represented by the external continuous manifold S1, while the higher inductive circle S1 gets interpreted as the constant sheaf at the homotopy type of the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the Yoneda lemma, then, any function definable on S1 is necessarily continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since functions f : X → Y in HoTT are defined simply by specifying an element f(x) : Y in the context of a free variable x : X, variation in a free variable confers a liminal sort of continuity: such an expression could be interpreted in a spatial ∞-topos in which case it necessarily defines a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Shulman’s spatial type theory works by introducing the notion of a crisp free variable to get around this liminal continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' An expression in spatial type theory depends on its crisp free variables discontinuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The modalities ♭ and ♯ of spatial type theory represent crisp variables universally on the left and right respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this way, ♭X is the discrete retopologization of the spatial type X, while ♯X is its codiscrete retopologization — a map out of ♭X is a discontinuous map out of X, while a map into ♯X is a discontinuous map into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Spatial type theory is intended to be interpreted into local geometric morphisms γ : E → S of ∞-toposes, those for which γ∗ has a fully faithful right adjoint γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' which gives a geometric morphism f : S → E (with f∗ := γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=') adjoint to γ which acts as the focal point of E as a space over S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The adjoint modalities ♭ and ♯ are interpreted as the adjoint idempotent (co)monad pair γ∗γ∗ and γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='γ∗ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A crisp free variable is then one which varies over an object of the focal point S : a free variable is crisp when it is in focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There is not only one way for mathematical objects to be spatial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Spaces may cohere with smooth, analytic, algebraic, condensed, and simplicial or cubical combinatorial structures — and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Each of these cases would give rise to a particular spatial type theory as the internal language of an appropriate local ∞-topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But there are many cases arising in practice where we need not just one axis of spatiality, but many at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, it is a classical theorem that the homotopy type of a manifold may be computed as the realization of a (topologically discrete) simplicial set associated to the ˇCech nerve of a good open cover of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This theorem relates a simplicial set to a continuous space, via an intermediary simplicial space which is both continuous and simplicial at the same time — the ˇCech nerve of the cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But in spatial homotopy type theory there is only one notion of crisp variable, and therefore just one sort of spatiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For simplicial types, the discrete reflection is the 0-skeleton sk0, while the codiscrete reflection is the 0- coskeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For simplicial spaces, we then have both the (topologically) discrete ♭ and codiscrete ♯, as well as the simplicially 0-skeletal sk0 and 0-coskeletal csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Interestingly, the ˇCech nerve itself arises from these modal- ities: the ˇCech nerve of a map f : X → Y between 0-skeletal types (that is, continuous or differential stacks with no simplicial structure) is its csk0-image, as we will see later in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A simplicial space has both a shape SX and a realization (or colimit) reX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' the first is a topologically discrete simplicial type, while the latter is a 0-skeletal but spatial type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' With all these modalities, we can prove the theorem about good covers described above as Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3 Another use case for multiple axes of spatiality is Sati and Schreiber’s Proper orbifold cohomology [43], where orbifolds are understood both as having both differential structure (as differential stacks) and global equivariant structure (concerning their singularities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In order to get the correct generalized cohomology of orbifolds without relying on ad-hoc constructions based on a global quotient presentation of the orbifold, Sati and Schreiber work with the ∞-topos of global equivariant differential stacks, which is local both over the global equivariant topos and the topos of differential stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Here the differential modalities S, ♭ and ♯ are augmented with the modalities of equivariant cohesion [41]: < , ⊂ , and ≺ , which take the strict quotient, the underlying space as an invariant type, and the Yoneda embedding of the underlying space of a global equivariant type respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Again the modalities play a central role in the theory, with the ordinary Borel cohomology of a global quotient orbifold X �G being the ordinary cohomology of S ⊂ (X �G), while the proper equivariant Bredon cohomology of X �G is the cohomology of ≺ (X �G), twisted by the map to ≺ BG classifying the quotient map ≺ X → ≺ (X �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In these cases, modalities that lie in the same position in their adjoint chain commute with each other, so, for example, ♭ commutes with sk0 and ♯ commutes with csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' However, there are cases where these modalities are nested, with one spatiality being a refinement of another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This occurs for example in supergeometry as formulated by Schreiber in [44] with the modalities of solid cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The supergeometric focus is given by the even comodal- ity ⇒ (which takes the even part of a superspace) and the rheonomic modality Rh which is given by localizing at the odd line R0|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this paper, we put forward a modification of spatial type theory to allow for multiple axes of spatiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our theory works by allowing for a meet semi-lattice of focuses ♥,♣,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=', each with a separate notion of ♥-crisp variable and pair of adjoint (co)modalities ♭♥ and ♯♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Like spatial type theory, our custom type theory gets us to the coalface of synthetic homotopy theory very efficiently while staying simple enough to be used in an informal style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The presence of multiple notions of crispness forces a more complex context structure than spatial type theory’s separation of the context into a crisp zone and cohesive zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Similar to many other modal type theories [30, 22, 21, 9, 38], we annotate each variable with modal information, here, the focuses for which that variable is crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The typing rules for the modalities of each focus then work essentially independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The exception is ♭-elimination, which is upgraded to allow the crispness of the term being eliminated to be maintained in the variable bound by the induction (a ‘crisp’ induction principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Ours is far from the only extension of type theory with multiple modalities, but as we discuss in more detail later, no existing theory has the combination of features that we are looking for: dependent types (ruling out [30]) that may depend on modal variables (ruling out [9]), multiple commuting comodalities (ruling out [47, 11, 38]) each with a with right-adjoint modality (ruling out [33]) and no further left-adjoints (ruling out [22, 21] and [16, §14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In addition to allowing us to formalize the theorem about ˇCech nerves of open covers as Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5, our type theory will be able to handle the equivariant differential cohesion used by Sati and Schreiber in their Proper orbifold cohomology [43], as well as the nested focuses of Schreiber’s supergeometric solid cohesion [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This extends the work of Cherubini [17] and the first author [34, 35, 36] of giving synthetic accounts of the constructions of Schreiber [44] and Sati-Schreiber [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Positing an additional focus does not disturb arguments made using existing focuses, so we also expect our theory to be helpful when dipping into simplicial arguments in the course of other reasoning by adding a simplicial focus and making use of the new modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The problem of defining simplicial types in ordinary Book HoTT remains open, and there are now a number of different approaches to constructing simplicial types which each use some extension to the underlying type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this paper, we will axiomatize the 1-simplex ∆[1] as a linear order with distinct top and bottom and use the cohesive modalities to define the ˇCech nerve of a map and the realization or colimit of a simplicial type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We believe our approach here would pair nicely with other approaches to simplicial types for the purposes of synthetic (∞,1)-category theory such as [42, 14, 49, 48], where the sk0 modality would take the core of a Rezk type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 1Though we have not looked in detail at how the focuses would work with the Riehl-Shulman simplicial type theory, and in particular how they would interact with the cubes/topes zones of the Riehl-Shulman context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4 Outline of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' After presenting our type theory in §2, we will look at ways to specialize the spatiality of a focus in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In particular, we will observe that in many cases there is a small class of test spaces Gi so that codiscreteness (that is, being ♯-modal) is detected by uniquely lifting against the ♭-counits ♭Gi → Gi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' such Gi will be said to detect continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Externally, the Gi could be any family which generates the logos under colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In practice, the Gi will be test spaces which minimally carry the appropriate spatiality: in the simplicial case, the simplices ∆[n];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' in the real-cohesive case, the Euclidean spaces Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' for condensed sets, the profinite sets, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In §3, we will also meet a family of axioms which hold for spatialities that are locally contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, continuous manifolds which are built from Euclidean spaces by colimits are locally contractible, while condensed sets which are built from profinite sets by colimits need not be locally contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In general, a space is locally contractible when it has a constant shape in the sense of Lurie [31, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may define a space C to be contractible when any map C → S to a discrete space S is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If the converse holds — a space S is discrete (♭-modal) if every map C → S is constant — then we say that C detects the connectivity of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, R detects the connectivity of continuous ∞-groupoids, and ∆[1] detects the connectivity of simplicial ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If there is a space (or family of spaces) which detects connectivity, then the local geometric morphism p corresponding to the morphism is furthermore strongly locally contractible in that p∗ has a left adjoint p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' which takes the (constant value of the) shape of a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the case that p is both local and strongly locally contractible, we say that p is cohesive following Lawvere [28], Schreiber [44], and Shulman [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Nullifying at the family of spaces which detect connectivity gives a modality S which is left adjoint to ♭;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' it may be thought of as taking the homotopy type of a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In §4 we will give example axioms for specializing single focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will review Shulman’s axioms for real cohesion, where the Euclidean spaces Rn detect continuity and connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will then see simplicial cohesion in some detail, where the simplices ∆[n] detect continuity and connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We give our types simplicial structure by axiomatizing the 1-simplex ∆[1] as a total order with distinct top and bottom elements, following Joyal’s characterization of simplicial sets as the classifying topos for such orders [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We use the csk0 modality to construct ˇCech nerves of maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then we will describe the global equivariant cohesion first observed by Rezk [41] and used by Sati and Schreiber in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we will briefly describe axioms for topological focuses such as Johnstone’s topological topos of sequential spaces [25] and the condensed/pyknotic topos of Clausen-Scholze [19] and Barwick-Haine [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' After surveying some of the different sorts of spatiality which types might carry, we turn our attention to multiple focuses in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3, we define what it means for two cohesions to be orthogonal: when the family which detects the connectivity of one is discrete with respect to the other, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We then prove a few lemmas concerning orthogonal cohesions, in particular concerning when it is possible to commute the various modalities past each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we give examples of multiple focuses in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We begin with simplicial real cohesion, which has both a simplicial focus and a real-cohesive focus which are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We prove, in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5, that the shape of any 0-skeletal type M may be computed as the realization of a topologically discrete simplicial type constructed from the ˇCech nerve of any good cover U of M — one for which finite intersections of the Ui are contractible in the sense of being S-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Next, we combine equivariant cohesion with differential cohesion to give the series of modalities used in Sati and Schreiber’s Proper orbifold cohomology [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Happily, no extra axioms are needed to show that the two cohesions are orthogonal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' we prove this in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we describe the supergeometric or “solid” cohesion of Schreiber’s Differential Cohomology in a Co- hesive ∞-topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This extends real cohesion with the odd line R0|1, where the “discrete” comodality of the super- geometric focus takes the even part of a supergeometric space, and the “codiscrete” modality takes a rheonomic reflection of the space, one whose super structure is uniquely determined by its even structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Unlike our other examples where the focuses involved are orthogonal, here the differential focus is included in the supergeometric focus: any discrete space is also purely even, as is any codiscrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We would like to thank Urs Schreiber for his careful reading and extensive comments during 5 the drafting process of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' And we would like to thank Hisham Sati for his feedback and words of encour- agement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The authors are grateful for the support of Tamkeen under the NYU Abu Dhabi Research Institute grant CG008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2 A Type Theory with Commuting Focuses The fundamental duality in higher topos theory is between the ∞-topos — a general sort of space — and the ∞- logos — the category of sheaves of homotopy types on such a space [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This duality is perfect: a map of ∞-toposes E → F is defined to be a lex accessible functor Sh∞(F) → Sh∞(E ) between their corresponding ∞-logoses in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This duality between toposes and logoses gives a nice perspective on the distinction between the petite toposes, which are used as generalized spaces in practice, and the gros toposes — or rather, their dual logoses — which are used as categories of spaces, rather than as spaces in their own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Quite opposite to their names, the petite toposes are “big” spaces, while the gros toposes are “small” spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' it is their dual logoses which are correctly described by the adjectives “petite” and “gros”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since the logos is the category of sheaves on the topos, or equivalently the category of ´etale maps into the topos, the “larger” the topos the more constraining the ´etale condition becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For that reason, the gros toposes have qualitatively “smaller” categories of sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On the other hand, the more general the ´etale spaces may be, the “smaller” the base topos must be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In general, the “biggest” logoses, the logoses of spaces, must correspond to the “smallest” toposes: those toposes which are infinitesimal patches around a focal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This point of view is emphasized in Chapter 4 of DCCT [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may therefore, as a first pass, identify logoses of spaces as dual to those toposes E which are local over a focal point F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A geometric morphism p : E → F is local when it admits a left adjoint right inverse f : F → E in the (∞,2)-category of toposes which we call the focal point of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If E is a topological space (that is, if its corresponding logos Sh∞(E ) is the category of sheaves Sh∞(X) on a sober topological space X), then the terminal geometric morphism γ : E → S is local just when X has a focal point: a point f ∈ X whose only open neighborhood is the whole of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In particular, the prime spectrum of a ring A is local if and only if A is a local ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' in this case, the focal point is the unique maximal ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On the logos side, this means that the direct image p∗ admits a fully faithful right adjoint p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' (which is f∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' All together, this gives an adjoint triple between the corresponding logoses: Sh∞(E ) Sh∞(F) p∗ p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' p∗ Thinking of the objects of Sh∞(E ) as generalized spaces and the objects of Sh∞(F) as mere homotopy types (sheaves on a point), we may see the direct image p∗ as taking the underlying homotopy type of points of a space, while p∗ and p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' are the discrete and codiscrete topologizations of bare homotopy types, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This adjoint triple gives rise to an adjoint pair p∗p∗ ⊣ p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='p∗ of a idempotent comonad p∗p∗ and idempotent monad p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='p∗ on the logos Sh∞(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Understood as operations on spaces, these are the discrete and codiscrete retopologizations of a space respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Examples of local toposes with focal point F having category of sheaves Sh∞(F) = ∞Grpd the ∞-category of ∞-groupoids include simplicial types ∞Grpd∆op (where discrete is 0-skeletal and codiscrete is 0-coskeletal),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' continuous and differentiable ∞-groupoids2 Sh∞({Rn}) (where discrete means all charts are constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' and codis- crete means that any function valued in the set of points is a chart),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' condensed ∞-groupoids (where discrete means discrete,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' and codiscrete means codiscrete),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' and global equivariant ∞-groupoids ∞GrpdGloop (where discrete means 2These are the gros toposes of C 0 and C ∞ manifolds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6 being a constant presheaf on the global orbit category, and codiscrete means being a presheaf representable by an ordinary ∞-groupoid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Shulman [46] has shown that every ∞-logos may be presented by a model of homotopy type theory, allowing reasoning conducted in homotopy type theory to be interpreted in any ∞-logos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this sense, homotopy type theory is to ∞-logoses as set theory is to the 1-logoses of Grothendieck, Lawvere, and Tierney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In Brouwer’s Fixed Point Theorem in Real-Cohesive Homotopy Type Theory [47], Shulman also put forward a spatial type theory which may (conjecturally) be interpreted into any local geometric morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Spatial type theory is characterized by including an adjoint pair ♭ ⊣ ♯ of a lex comodality ♭ and lex modality ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These are to be interpreted as p∗p∗ and p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='p∗ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In spatial type theory, any type has a spatial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The existence of this spatial structure is witnessed by the two opposite ways that we can get rid of it: either we can remove all the spatial relationships between points, using the “discrete” ♭ comodality, or we can trivialize the spatial relations using the “codiscrete” ♯ modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We emphasize that this spatial structure is distinct from the homotopical structure that all types have by virtue of the identifications between their elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, the topological circle S1 := {(x,y) : R2 | x2 +y2 = 1} has a spatial structure as a subset of the Euclidean plane (as a sheaf on the site of continuous manifolds, for example), but is a homotopy 0-type (or “set”) without any non-trivial identifications between its points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' in particular ΩS1 = ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The homotopy type S1 of the circle, however, is spatially discrete but has many non-trivial identifications of its point: in particular ΩS1 = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There is not, however, only one way to be spatial in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example a simplicial topological space has both a simplicial structure and a topological structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This can be witnessed at the level of toposes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If p : E → F admits a focal point f : F → E , then f ∆op : F ∆op → E ∆op is also a focal point of p∆op : E ∆op → F ∆op, where the logos Sh∞(E ∆op) := (Sh∞(E ))∆op is the category of simplicial objects in the logos Sh∞(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But there is another local geometric morphism γ : E ∆op → E where γ∗ sends a simplicial sheaf X• to X0 and γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' is given by the 0-coskeleton csk0 Sn := Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These two different axes of spatiality on the objects of Sh∞(E )∆op commute, in that the following diagram of adjoints commutes: Sh∞(E )∆op Sh∞(F)∆op Sh∞(E ) Sh∞(F) p∆op ∗ γ∗ γ∗ p∗ In particular, we have that p∗∆op p∗∆op and γ∗γ∗ commute as endofunctors of Sh∞(E )∆op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The former discretely retopologizes a simplicial space, while the latter includes the space of 0-simplices as a 0-skeletal simplicial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Each focus gives an axis along which the objects of the top logos Sh∞(E )∆op may carry spatial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' When working in, say, simplicial differential spaces, we would like to have access to both the S ⊣ ♭ ⊣ ♯ of real cohesion and the re ⊣ sk0 ⊣ csk0 of simplicial cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Shulman’s spatial type theory offers no way to do this: the ♭ and sk0 comonads have incompatible claims on the notion of ‘crisp’ variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The solution is to allow a separate notion of crispness for each focus we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this section, we will describe the rules for a type theory with commuting focuses, generalizing ordinary spatial type theory in the case of a single non-trivial focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will then describe axioms which make these into commuting cohesions, in the sense of cohesive type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To this end, we will fix an commutative idempotent monoid Focus of focuses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' we will write the product of the focus ♥ and the focus ♣ as ♥♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This product induces an ordering on the focuses by saying that ♣ ≤ ♥ whenever 7 ♣♥ = ♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' With respect to this ordering, the product becomes the meet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' we may therefore also think of Focus as a meet semi-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will write the identity element of Focus as ⊤, and note that it is the top focus with respect to the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For most of our purposes in this paper, our commutative idempotent monoid Focus of focuses will be freely generated by a finite set of basic focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Explicitly, we may take Focus = Pf (BasicFocuses)op to be the set of finite subsets of the set of basic focuses with union as the product, and therefore the opposite of the ordering of subsets by inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' All variables in the context will be annotated with the focus that they are in: x :♥ X ⊢ t : T In general, we will abbreviate the context entry x :⊤ X as x : X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the case that Focus = {♥ ≤ ⊤} is freely generated by one basic focus, we recover the split context used in Shulman’s spatial type theory, where our context x :♥ X, y :⊤ Y ctx corresponds to Shulman’s context x :: X | y : Y ctx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To describe the typing rules, we will need a couple of auxiliary operations on contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The first operation ♥Γ adds a specific focus ♥ to the annotation on every variable in a context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' So: ♥(·) :≡ · ♥(Γ,x :♣ A) :≡ (♥Γ),x :♥♣ A We also need an operation ♥ \\ Γ that deletes any variables not contained within a given focus ♥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' this is the equivalent of going from ∆ | Γ ctx to ∆ | · ctx in ordinary spatial type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♥\\(·) :≡ · ♥\\(Γ,x :♣ A) :≡ � (♥\\Γ),x :♣ A if ♣ ≤ ♥ ♥\\Γ otherwise We say that a variable x :♣ X is ♥-crisp if ♣ ≤ ♥, and so the ♥-crisp variables are precisely those that survive the ♥\\Γ operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We say that a term t : T is ♥-crisp if both it and its type T only contain ♥-crisp variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=', it is well-formed in context ♥\\Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We are now ready to describe the rules of the type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' All the usual type formers — Σs, Πs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' — will be included as usual, only referring to variables of the top focus ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the convention that x :⊤ X be written as x : X, these rules look exactly as they do usually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We therefore focus on the new features of type theory with commuting focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Structural Rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' CTX-EMPTY · ctx CTX-EXT ♥\\Γ ⊢ A type Γ,x :♥ A ctx VAR Γ,x :♥ A,Γ′ ctx Γ,x :♥ A,Γ′ ⊢ x : A In prose, these rules read as follows: CTX-EMPTY: The empty context is a context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' CTX-EXT: If A is a ♥-crisp type in context Γ, then Γ,x :♥ A ctx is a context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' VAR: If x :♥ A appears in a context, then the variable x has type A in that context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Given a context Γ,x :♥ A ctx, it must be the case that A only depends on the variables in Γ which are themselves ♥-crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This careful context formation rule is what replaces the division of the context into two zones in Shulman’s spatial type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the conclusion of the variable rule, the type A is well-formed in context Γ,x :♥ A,Γ′ ctx by the admissible DIVIDE-WK rule given below, followed by further weakening with Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 8 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Rather than annotating variables, may be tempting to try a floating context separator |♥ for each focus, so that the variables to the left of |♥ are precisely the ♥-crisp ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Such contexts are not sufficiently general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' specifically, the ♭-elimination rule will let us produce a context containing x :♥ A,y :♣ B which clearly cannot be separated in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The following rules and equations will be made admissible, with the proofs sketched in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' WK Γ,Γ′ ⊢J Γ,x :♥ A,Γ′ ⊢J −−−−−−−−−− SUBST ♥\\Γ ⊢ a : A Γ,x :♥ A,Γ′ ⊢J Γ,Γ′[a/x] ⊢J [a/x] −−−−−−−−−−−−−−−−−−−−− PROMOTE-CTX Γ ctx ♥Γ ctx −−−− PROMOTE Γ ⊢J ♥Γ ⊢J −−−−− DIVIDE-CTX Γ ctx ♥\\Γ ctx −−−−−− DIVIDE-WK ♥\\Γ ⊢J Γ ⊢J −−−−−− ♥(♣Γ) ≡ (♥♣)Γ ♥\\(♣\\Γ) ≡ (♣♥)\\Γ First, we have ordinary weakening by a variable, and a ‘crisp’ substitution similar to that used in spatial type theory, where crisp variables may only be substituted with similarly crisp terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These specialize to the ordinary weakening and substitution rules when used for ♥ ≡ ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' PROMOTE-CTX corresponds to the application of the endofunctor ♥ to the context Γ, and PROMOTE to precomposition with the counit morphism ♥Γ → Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DIVIDE-CTX gives the largest ‘subcontext’ ♥ \\ Γ of Γ such that there is a substitution Γ → ♥(♥ \\ Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The context operation ♥ \\ − thus acts like a left-adjoint to ♥−, although semantically a left-adjoint may not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Rules for ♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We now come to the rules for the ♭ comodality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-FORM ♥\\Γ ⊢ A type Γ ⊢ ♭♥A type ♭-INTRO ♥\\Γ ⊢ M : A Γ ⊢ M♭♥ : ♭♥A ♭-ELIM ♣♥\\Γ ⊢ A type Γ,x :♣ ♭♥A ⊢ C type ♣\\Γ ⊢ M : ♭♥A Γ,u :♣♥ A ⊢ N : C[u♭♥/x] Γ ⊢ (let u♭♥ := M inN) : C[M/x] ♭-BETA ♣♥\\Γ ⊢ A type Γ,x :♣ ♭♥A ⊢ C type ♣♥\\Γ ⊢ K : A Γ,u :♣♥ A ⊢ N : C[u♭♥/x] Γ ⊢ (let u♭♥ := K♭♥ inN) ≡ N[K/u] : C[K♭♥/x] In prose, these rules read as follows: ♭-FORM: If A is a ♥-crisp type, then we may form ♭♥A type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-INTRO: If M is a ♥-crisp term of type A, then we may form M♭♥ of type ♭♥A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-ELIM: If C is a type depending on the ♣-crisp variable x :♣ ♭♥A, and M : ♭♥A is a ♣-crisp element of type ♭♥A, then we may assume that M is of the form u♭♥ for a ♣♥-crisp variable u :♣♥ A when defining an 9 element of C[M/x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We write this element as (let u♭♥ := M inN) : C[M/x] where N : C[u♭♥/x] is the element we defined assuming that M was of the form u♭♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The equation ♥\\(♣\\Γ) ≡ (♣♥)\\Γ is necessary here to know that the type ♭♥A is well-formed in context ♣\\Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-BETA: If M actually is of the form K♭♥ for suitably crisp K, then we simply substitute K in for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The term K must be ♣♥-crisp for both the ♭-INTRO and ♭-ELIM to have been applied, and so its substitution for the ♣♥-crisp variable u is well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These rules are stronger than the ones used by Shulman for spatial type theory, even in the case of a single focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We have built in a ♣-crisp induction principle for ♭♥, for any two focuses ♥ and ♣: if the term we are inducting on is already ♣-crisp, then we may maintain that crispness in the new assumption u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If we have a single non-trivial focus ♥, as is the case in Shulman’s type theory, then taking ♣ = ♥ in the above expression yields the ‘crisp ♭ induction’ principle of [47, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This induction principle is proven by taking a detour through ♯, but here we choose to build it into the rule directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our elimination rule is in fact also admissible from the less general one that requires the freshly bound variable to only be ♥-crisp, but we choose the more general rule for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Rules for ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The rules for ♯ are a little simpler, and in the case of a single focus specialize exactly to the rules of spatial type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-FORM ♥Γ ⊢ A type Γ ⊢ ♯♥A type ♯-INTRO ♥Γ ⊢ M : A Γ ⊢ M♯♥ : ♯♥A ♯-ELIM ♥\\Γ ⊢ N : ♯♥A Γ ⊢ N♯♥ : A ♯-BETA ♥\\Γ ⊢ M : A Γ ⊢ (M♯♥)♯♥ ≡ M : A ♯-ETA Γ ⊢ N : ♯♥A Γ ⊢ N ≡ (N♯♥)♯♥ : ♯♥A In prose, these rules read as follows: ♯-FORM: When forming the type ♯♥A, all variables may be used in A as though they are ♥-crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-INTRO: When forming a term M♯♥ : ♯♥A, all variables may be used in M as though they are ♥-crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-ELIM: If N is a ♥-crisp element of ♯♥A, we may extract an element N♯♥ : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-BETA: If M is a ♥-crisp element of A, then M♯♥♯♥ ≡ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-ETA: Any term of N : ♯♥A is definitionally equal to N♯♥ ♯♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As in ordinary spatial type theory, the term N♯♥ may not be well-typed on its own, because it may use non-crisp variables of the context Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is however well-typed underneath the outer (−)♯♥, since the introduction rule allows us to use any variable as though it is ♥-crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Perhaps surprisingly, the shape of the ♯-FORM and ♯-INTRO rules is what builds the left-exactness of ♭ into the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is the case even in ordinary spatial type theory, not a feature that only appears in this multi-focus setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The trick is that the promotion operation ♥Γ distributes over the context extensions in Γ rather than being a ‘stuck’ context former applied to Γ as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Specifically, when using ♯ to derive crisp Id-induction, one applies ♯ to a type x :: A,y :: A, p :: (x = y) | · ⊢ C type, yielding a type | x : A,y : A, p : (x = y) ⊢ ♯C type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Internalized, the former context represents the type (x : ♭A)×(y : ♭A)×♭(x♭ = y♭), but ♯-FORM treats it as identical to ♭((x : A)×(y : A)×(x = y)) when applying adjointness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 10 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In most cases of interest, our commutative idempotent monoid of focuses is freely generated by a finite set of basic focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this situation, it suffices to provide the ♭ and ♯ only for the basic focuses, as the remainder can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The top focus ⊤ (which semantically corresponds to the entire topos we are working in) has both ♭⊤A and ♯⊤A canonically equivalent to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' And given focuses ♥ and ♣, it is quickly proven that ♭♥♣ is equivalent to ♭♥♭♣ and similarly for the ♯s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Related Type Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Besides the original spatial type theory, there are several other dependent modal type theories that come close to our needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The ‘adjoint type theory’ perspective [40, 29, 30] was the guiding principle that led to the original spatial type theory of [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Indeed, when instantiated with appropriate mode theory, the framework of [30] reproduces a simply typed version of the theory presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The specific mode theory to be used is a cartesian monoid with a system of commuting, product-preserving endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A dependently typed variant of adjoint type theory is not yet forthcoming, but we expect that our dependent type theory would be an instance of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' An separate line of work on modal type theories is Multimodal Type Theory [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In MTT, every mode morphism µ is reified in the type theory as a positive type former, and each modality modµ must have a left- adjoint-like context operator written �µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If we do not assume the existence of S, then we are only able to describe ♯ in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Later work [21] describes a multimodal type theory where each mode morphism becomes a (more convenient) negative type former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The semantic requirements are even stronger: the functor corresponding to the modality must be a dependent right-adjoint [11], whose left adjoint is itself a parametric right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is too strict even to capture ♯ without additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In [16, §14], an alternative ‘cohesive type theory’ is presented, using a combination of the above two styles of modal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Rather than working with the endofunctors on the topos of interest, the cohesive setting is kept as an adjoint quadruple Π0 ⊣ Disc ⊣ Γ ⊣ CoDisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A positive type former is used for Disc and negative type formers for Γ and CoDisc, due to the requirements on having one or two left-adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is likely that this could be extended to commuting cohesions, but the interactions of the various context �− operations for the left-adjoints may be difficult to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The type theory with context structure most formally similar to ours is ParamDTT [38, 37], where variables in the context annotated with a modality indicate a variable under that modality directly, not its left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is from this work that we take the left-division notation − \\ Γ for the clearing operation on contexts, which itself has appeared in other guises, for example [39, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ParamDTT uses a fixed ‘mode theory’ with three modalities {¶,id,♯} equipped with a particular composition law, but it is clear that the rules for contexts and basic type formers would work equally well for other sets of modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A version of the cohesive ♭ can be derived from the ‘modal Σ-type’, fixing the second component to be the unit type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There does not appear to be a way to derive the ordinary (negative) rules for ♯ in ParamDTT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3 Specializing a Focus A focus gives a specific axis along which a type may be spatial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In simplicial cohesion, we have a simplicial focus sk0 ⊣ csk0 and in differential cohesion a differential focus ♭ ⊣ ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But what makes the simplicial focus simplicial and the differential focus differential?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this section, we will investigate two axioms schemes which can determine the peculiarities of a given focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the next section, we will see these axioms in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First, we note that with a single focus, type theory with commuting focuses is the same theory as Shulman’s spatial type theory in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Any of the lemmas and theorems proven in §3, 4, 5, and 6 of Real Cohesion [47] concerning ♭ and ♯ and using no axioms are true also of ♭♥ and ♯♥ for any fixed focus ♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorems which do involve the use of axioms are also valid, so long as the crispness used in those axioms is interpreted as ♥-crispness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The rules for ♭♥ and ♯♥ specialize to Shulman’s rules, and therefore his proofs carry through directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 11 Specifically, ♭♥ is a coreflector and ♯♥ is a monadic modality, both are lex, and ♭♥ is (♥-crisply) left-adjoint to ♯♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since adding a focus only expands the rules of the type theory and does not restrict the application of any of the rules for any of the other focuses, any of the theorems proven in this section for a single focus will apply when working with multiple focuses as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For the rest of this section, we will work within a single focus ♥, and for that reason we will drop the anno- tations by ♥ in our expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, we will write ♭♥ as simply ♭, and we will write x :♥ X as x :: X, following Shulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 Detecting Continuity In this section, we will look at an axiom which ties the liminal sort of “continuity” implied by the crisp variables of the type theory to the concrete continuity of a particular type G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our axiom will take the form of a lifting property characterizing those crisp maps which are ♯-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As we will show in the upcoming Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2, a crisp map is ♯-modal if and only if it lifts crisply (in a sense made precise in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1) against all of the ♭-counits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let c :: A → B and f :: X → Y be crisp maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We say that c lifts crisply against f if for any crisp square as on the left below, there is a unique crisp filler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A X B Y c f ∀ ∀ ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭(XB) ♭(XA) ♭(Y B) ♭(Y A) ♭(◦ f) ♭(c◦) ♭(◦f) ♭(c◦) ⌟ More formally, we write c ⊥♭ f for the proposition that the square on the right is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A crisp map f :: X → Y is ♯-modal if and only if for all crisp A, (ε : ♭A → A) ⊥♭ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If f is ♯-modal, then since ♯ is lex, it lifts on the right against all ♯-equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For any crisp A, the ♭-counit ε : ♭A → A is a ♯-equivalence by [47, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, the square XA X♭A Y A Y ♭A f◦ ε f◦ ε ⌟ is a pullback, and since ♭ preserves crisp pullbacks ([47, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='10]), we see that ε ⊥♭ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On the other hand, suppose that f lifts crisply on the right against all ♭-counits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To show that f is ♯-modal, it will suffice to show that its ♯-naturality square is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let X → ♯X ×♯Y Y be the gap map of the ♯-naturality square of f, seeking to show that this map is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It suffices to split the gap map over the naturality square, by the universal property of the pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' So, consider the crisp square ♭(♯X ×♯Y Y) X ♯X ×♯Y Y Y snd ε f F k where F(t) :≡ (let t := p♭ in(fst p)♯) is a version of the first projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To check that the square commutes, it suffices by ♭-induction to give, for crisp elements u :: ♯X, y :: Y, and p :: (♯f(u) = y♯), a term of type f(u♯) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But we have crisply that ♯f(u) ≡ ♯f(u♯♯) = (f(u♯))♯ by the definition of ♯f, and composing this path with p we know 12 p′ :: (f(u♯))♯ = y♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the lexness of ♯, we therefore also have p′′ :: ♯(f(u♯) = y), so that the square commutes by p′′♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By hypothesis, there is a unique crisp map k : ♯X ×♯Y Y → X filling this square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The bottom triangle says precisely that k lives over the second projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will turn the top triangle into a proof that k lives over the first projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let (u,y, p) : ♯X ×♯Y Y, seeking to show that k(u,y, p)♯ = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This latter type of paths is codiscrete (because ♯X is codiscrete), and so when mapping into it we may assume by that u is of the form x♯, reducing our goal to k(x♯,y, p)♯ = x♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the lexness of ♯, it suffices to give an element of ♯(k(x♯,y, p) = x), and for this it suffices to give an element of k(x♯,y, p) = x under the hypotheses that x, y, and p are crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this case, (x♯,y, p)♭ : ♭(♯X ×♯Y Y), and so we have that k(x♯,y, p) = F((x♯,y, p)♭) by the upper triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But by definition, F((x♯,y, p)♭) ≡ x♯♯ ≡ x, so that we have succeeded in giving the desired identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We have shown that k lives over the naturality square;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' now we need to show that it splits the gap map X → ♯X ×♯Y Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To that end, consider the following diagram: ♭X ♭(♯X ×♯Y Y) X X ♯X ×♯Y Y Y gap ♭gap ε ε f f k Showing that the diagram commutes as drawn follows easily by ♭-induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We then have two crisp fillers of the outer square: first we have idX : X → X and k ◦gap : X → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the uniqueness of crisp fillers, we conclude that these must be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Knowing that the crisp ♯-modal maps may be characterized by lifting crisply against ♭-counits suggests that we could axiomatize the particular qualities of ♯ by restricting the class of ♭-counits which it suffices to lift against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To that end, we make the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let G :: I → Type be a crisp type family indexed by a ♭-modal and inhabited type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We say that G detects continuity when, for every crisp map f :: X → Y, { f is ♯-modal} �(ε : ♭Gi → Gi) ⊥♭ f for all i :: I � Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Thinking externally, it is straightforward to see that any family Gi which generates a local topos E in question under colimits will detect continuity for the focus given by the terminal map of toposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is because ♭, as a left adjoint, commutes with all colimits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' therefore the problem of lifting against the ♭-counit of any object of E can be reduced to that of lifting against the ♭-counits of the generators Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A crisp type X is ♯-modal if and only if ♭(A → X) → ♭(♭A → X) is an equivalence for all crisp types A, and if G detects continuity then it suffices to check for each Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' When f : X → 1, the square defining crisp lifting is a pullback iff the top map is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If a family detects continuity, then it is a separating family for crisp maps in the following precise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that G :: I → Type detects continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f :: X → Y be a crisp map for which ♭f : ♭X → ♭Y is an equivalence and for all i :: I, the induced map ♭(Gi → X) → ♭(Gi → Y) given by post-composing with f is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then f is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First, note that f is a ♯-equivalence since it is by hypothesis a ♭-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It therefore suffices to show that f is ♯-modal, which by the assumption that G detects continuity means showing that f lifts crisply against all ♭-counits ♭Gi → Gi for i :: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Consider the following diagram: ♭(XGi) ♭(X♭Gi) ♭(♯XGi) ♭(Y Gi) ♭(Y ♭Gi) ♭(♯Y Gi) ♭(f◦) ♭(f◦) ∼ ∼ ♭(♯ f◦) The square on the left is the one we need to show is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For this, it will suffice to show that the middle map ♭(f◦) : ♭(X♭Gi) → ♭(Y ♭Gi) is an equivalence, since the leftmost vertical map is an equivalence by hypothesis and any square with two sides equivalences is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For that, the middle vertical map is equivalent by the adjunction ♭ ⊣ ♯ to the rightmost vertical map ([47, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But the rightmost vertical map is an equivalence because it is post-composition by the equivalence ♯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 Detecting Connectivity A focus is said to be cohesive if ♭ has a further left adjoint S which is itself a modality: ♭(SA → X) ≃ ♭(A → ♭X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This adjunction only determines S for crisp types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is better to define S by nullifying a family of objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' then S is determined for all types (of any size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To this end, we make the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let G :: I → Type be a crisp type family indexed by a ♭-modal type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We say that G detects connectivity when, for any crisp type X, {X is ♭-modal} �X is Gi-null for all i :: I � If G detects connectivity, then S is defined to be nullification at the family Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In Real Cohesion [47], the assertion that a given family G detects connectivity is known as Axiom C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In [44, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='48], a single object with this property is said to ‘exhibit the cohesion’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If there is a family G which detects connectivity, then we say that the focus is cohesive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is justified by the following theorem, which we may import directly from Real Cohesion [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that G detects connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then a crisp type is S-modal if and only if it is ♭-modal, and furthermore S is crisply left-adjoint to ♭: ♭(SA → X) → ♭(A → ♭X) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is [47, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4 Examples of Focuses To keep the various operators visually distinct, we will use completely different symbols for each focus we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The rules governing the type formers are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' S ⊣ ♭ ⊣ ♯ denotes real cohesion, where a set of real numbers (possible the Dedekind reals or an axiomatically asserted set of “smooth reals”) detects connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 14 re ⊣ sk0 ⊣ csk0 denotes simplicial cohesion, where the (axiomatically asserted) 1-simplex ∆[1] detects con- nectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' < ⊣ ⊂ ⊣ ≺ denotes global equivariant cohesion, where connectivity is detected by ≺ BG for finite groups G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This notational convention follows Sati and Schreiber [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Various topological toposes exhibit spatial type theory, with ♭ ⊣ ♯ retopologizing types with the discrete and codiscrete topologies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In particular, Johnstone’s topological topos has a focus whose continuity is detected by the walking convergent sequence N∞, which may be constructed as the set of monotone functions N → {0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 Real Cohesions In Real Cohesion [47], Shulman gives the axiom R♭ which states that a crisp type is ♭-modal if and only if it is RD-null, where RD is the set of Dedekind cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the terminology of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1, this says that RD detects continuous real connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Axiom 1 (Continuous Real Cohesion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume that RD detects continuous real connectivity, and also Shul- man’s Axiom T: For every x : RD, the proposition (x > 0) is ♯-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Though Shulman does not consider this axiom, we may also add the assumption that the family Rn D detects continuous real continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Using this assumption, we may internalize the arguments of Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='33 of [47] to show that the mysterious Axiom T follows from the proposition that if f :: Rn D → RD is crisp and f(x) > 0 for any crisp x :: Rn D, then in fact f(x) > 0 for all (not necessarily crisp) x : Rn D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since, by Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='28 of [47] (assuming the crisp LEM or the axiom of countable choice), any crisp Dedekind real is a Cauchy real, we are equivalently asking if a function f : RD → RD is positive on all Cauchy reals, is it always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This seems obvious, but as Shulman notes in Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='34, this obvious statement is not always true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' though assuming countable choice it is likely provable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There are continuous but non-differentiable functions f : RD → RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If we want to work in a topos where the types have a smooth structure instead of just a continuous structure, then we must work with a type of smooth reals RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The most common way to axiomatize the type of smooth reals is using the Kock-Lawvere axiom and the other axioms of synthetic differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 of [36] for a list of these axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In any case, if RS is a type of smooth reals, then we will take differential cohesion to mean that RS detects connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Axiom 2 (Differential Real Cohesion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If RS is a type of smooth reals (say, from synthetic differential geometry), then we assume that RS detects differential connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 Simplicial Cohesion There is a well known difficulty in describing simplicial types in ordinary homotopy type theory — the infinite amount of coherence data is difficult to describe formally given the tools of type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This difficulty has led to extensions of type theory such as two-level type theories [5, 8, 4] which augment HoTT with strict equalities which can be use to define simplicial homotopy types satisfying the simplicial identities strictly, bypassing the problematic tower of coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Another approach to avoiding simplicial difficulties is to simply interpret type theory into a topos of simplicial homotopy types, rather than mere homotopy types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is the approach taken by Riehl and Shulman in [42], where they present a type theory that makes every type into a simplicial type and has as primitives the simplices ∆[n], so that a simplex in a type A is a function σ : ∆[n] → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this section we will also work with simplicial homotopy types, but by different means to Riehl and Shulman’s type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Instead, we will describe simplicial cohesion, with adjoint modalities re ⊣ sk0 ⊣ csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These are defined semantically as follows: 15 The (“simplicial flat”) 0-skeletal comodality X �→ sk0 X sends a simplicial type to its 0-skeleton: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X2 X1 X0 sk0 �−−−−−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X0 X0 X0 The (“simplicial sharp”) 0-coskeletal modality X �→ csk0 X sends a simplicial type to its 0-coskeleton: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X2 X1 X0 csk0 �−−−−−−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X0 ×X0 ×X0 X0 ×X0 X0 The (“simplicial shape”) realization modality X �→ reX sends a simplicial type to its realization (or homotopy colimit), considered as a 0-skeletal simplicial type: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X2 X1 X0 re· �−−−−−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' colimXn colimXn colimXn Because simplicial sets are the classifying (0-)topos for strict intervals (totally ordered sets with distinct top and bottom elements) [50], and since the ∞-topos of simplicial homotopy types is the enveloping ∞-topos3 of simplicial sets [6], we may assume the existence of an interval ∆[1] to make sure that our type theory is interpreted in an ∞-topos equipped with a geometric morphism to S ∆op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may then define the n-simplex ∆[n] to be the n-length increasing sequences in ∆[1], and define an n-simplex in a type X to be a map ∆[n] → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Axiom 3 (Simplicial Axioms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We presume that ∆[1] is a total order with distinct top and bottom elements which we call 1 and 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Explicitly, this means that we have elements 0, 1 : ∆[1] and a proposition x ≤ y : Prop for x, y : ∆[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This order must satisfy the following axioms: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For all x, x ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For all x, y, and z, if x ≤ y and y ≤ z then x ≤ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For all x and y, if x ≤ y and y ≤ x, then x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For all x, y, either x ≤ y or y ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For all x, 0 ≤ x and x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 0 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3The enveloping ∞-topos of a topos is its free (homotopy) cocompletion, fixing existing homotopy colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 16 From these axioms, we may define the other simplices ∆[n] to be the chains of length n in ∆[1]: ∆[n] :≡ {⃗x : ∆[1]n | x1 ≤ x2 ≤ ··· ≤ xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We also assume the following: (Axiom ∆sk0) ∆[1] detects simplicial connectivity: a simplicially crisp type X is 0-skeletal if and only if every map ∆[1] → X is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The family ∆[−] : N → Type detects simplicial continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' (Axiom ∂∆) For i : ∆[1], we have csk0((i = 0)∨(i = 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Each ∆[n] is crisply projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' That is, for a simplicially crisp E : ∆[n] → Type, we have a map sk0((i : ∆[n]) → ∃Ei) → ∃sk0((i : ∆[n]) → Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As there is an obvious map the other way, this map is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let us quickly set the stage by proving that the n-simplices have trivial geometric realization and the 0-skeleton of the n-simplex ∆[n] is the ordinal [n] ≡ {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=',n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The order ∆[1] has finite meets and joins, and they distribute over each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Moreover, the inclusion {0,1} �→ ∆[1] is an inclusion of lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that x,y : ∆[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then either x ≤ y or y ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the former case, define x∧y :≡ x and x∨y :≡ y, and in the latter case x∧y :≡ y and x∨y :≡ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If both hold, then x = y and the definitions agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If x,y,z : ∆[1], then these three may find themselves in any of 6 orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' One may then check that in each of these cases, meets distribute over joins and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, supposing that x ≤ y ≤ z, then x ∧ (y ∨ z) = x∧z = x, while (x∧y)∨(x∧z) = x∨x = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The n-simplex ∆[n] is a retract of the n-cube ∆[1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Moreover, the inclusion [n] �→ ∆[n] given by i �→ 0 ≤ ··· ≤ 0 ≤ i times � �� � 1 ≤ ··· ≤ 1 is a retract of the inclusion {0,1}n → ∆[1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Given x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=',xn : ∆[1], define m1 :≡ � i:n xi and let i1 : n be its index, then m2 :≡ � n\\{m1} xi, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Note that m1 ≤ m2 ≤ ··· ≤ mn, so that m : ∆[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, if the xi were already in increasing order, then mi = xi, showing that this is a retract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We note that this retract argument works just as well on {0,1}n → [n], if we identify [n] with the subset {⃗x : {0,1}n | x1 ≤ ··· ≤ xn} of increasing sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since it only makes use of the lattice structure of {0,1}n and ∆[1]n, and the inclusion is a lattice homomorphism, we conclude that the necessary squares commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The n-simplex has trivial realization: re∆[n] = ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The realization re∆[n] is a retract of the realization re∆[1]n, and this is contractible since ∆[1] detects simplicial connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The inclusion [n] �→ ∆[n] given by i �→ 0 ≤ ··· ≤ 0 ≤ i times � �� � 1 ≤ ··· ≤ 1 is a sk0-equivalence, showing that sk0 ∆[n] ≃ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will show that [1] �→ ∆[1] is a sk0-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This will imply that [1]n �→ ∆[1]n is a sk0-equivalence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' since [n] �→ ∆[n] is a retract of this, we may conclude that it is a sk0-equivalence as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since this inclusion {0,1} �→ ∆[1] is simplicially crisp, to show that it is a sk0-counit it will suffice to show that it is a csk0-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We therefore need an inverse csk0 ∆[1] → csk0{0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since the codomain is 0-coskeletal, it suffices to define this map on ∆[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' So let i : ∆[1], seeking csk0{0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Axiom ∂∆, we have csk0((i = 0)∨(i = 1)), and since our goal is 0-coskeletal, we may assume that i = 0 or i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If i = 0, then we send it to 0csk0, if i = 1, then we send it to 1csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To show that this map is the inverse of csk0{0,1} → csk0 ∆[1], we may appeal to the fact that identities in a modal type are modal, and so we may remove the csk0 around csk0((i = 0) ∨ (i = 1)) and check that the maps invert each other on these elements, which they clearly do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We can also define the type of n-simplices in a simplicially crisp type, and prove a few elementary lemmas concerning the n-simplices of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let X be a simplicially crisp type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then define the type Xn of n-simplices in X as Xn :≡ sk0(∆[n] → X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If f : X → Y is a simplicially crisp map, then it induces a map fn : Xn → Yn by post-composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f : X → Y be a simplicially crisp map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If fn : Xn → Yn is an equivalence for all n, then f is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is a special case of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6, noting that X0 ≃ sk0 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f : X → Y be a simplicially crisp map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then for a crisp y : ∆[n] → Y, we have fib fn(ysk0) ≃ sk0((i : ∆[n]) → fib f (y(i))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We compute: fib fn(ysk0) ≡ (x : Xn)×(fnx = ysk0) ≡ (x : sk0(∆[n] → X))×let τsk0 := xin(f ◦τ)sk0 = ysk0 ≃ (x : sk0(∆[n] → X))×let τsk0 := xinsk0(f ◦τ = y) ≃ sk0((x : ∆[n] → X)×(f ◦x = y)) ≃ sk0((x : ∆[n] → X)×((i : ∆[n]) → (f(x(i)) = y(i)))) ≃ sk0((i : ∆[n]) → fib f (y(i))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f : X → Y be a simplicially crisp map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then (im f)n ≃ im fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We use the projectivity of the simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4 (im f)n ≡ sk0(∆[n] → (y : Y)×∃fib f (y)) ≃ (y : Yn)×let σsk0 := yinsk0((i : ∆[n]) → ∃fib f (σi)) ≃ (y : Yn)×let σsk0 := yin∃sk0((i : ∆[n]) → fib f (σi)) ≃ (y : Yn)×∃fib fn(y) ≡ im fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4This lemma is in fact equivalent to assuming the projectivity of the simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 18 The definition of the n-simplices that we gave above is simple, but it is not that straightforward to see that it is functorial in the ordinal [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We can give an alternative definition of the n-simplices which makes the functoriality evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let Interval denote the category of intervals: totally ordered sets with distinct top and bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The maps of Interval are the monotone functions preserving top and bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let FinOrd+ denote the category of finite inhabited ordinals and order preserving maps between them — the usual “simplex category”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We denote by [n] the ordinal {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=',n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will need a standard reformulation of the category of finite ordinals in terms of intervals (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [32, §VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7]) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There is a contravariant, fully faithful functor ι : FinOrdop + → Interval sending [n] to [n+1] with top element n+1 and bottom element 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To a map f : [n] → [m], we define ι f : [m+1] → [n+1] by ι f(i) :≡ � min{ j | i ≤ f(j)} n+1 if no such minimum exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Conversely, to a monotone map g : [m+1] → [n+1] preserving top and bottom, we define ι−1g : [n] → [m] by the dual formula (ι−1g)( j) :≡ max{i | g(j) ≤ i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may now define the n-simplices in a way which makes clear their functoriality in the category of finite inhabited ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We define the n-simplex ∆[n] to be ∆[n] :≡ Interval(ι[n],∆[1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, ∆ : FinOrd+ → Set gives a functor from finite inhabited ordinals to the category of sets, where ∆(f) : ∆[n] → ∆[m] is given by precomposing with ι f : [m+1] → [n+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Noting that [n] ∼= Interval(ι[n],[1]) by the fully-faithfulness of ι, the inclusion of top and bottom elements [1] �→ ∆[1] induces a natural inclusion [n] �→ ∆[n] by post-composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As we saw in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4, these inclusions are sk0-counits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 The ˇCech Complex The 0-coskeleton modality csk0 is useful for working in simplicial cohesion since it enables us to give an easy construction of the ˇCech complex of a map f : X → Y between 0-skeletal types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The ˇCech complex of such a map is, externally speaking, the simplicial type formed by repeatedly pulling back f along itself: ˇC(f) :≡ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X ×Y X ×Y X X ×Y X X Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f : X → Y be a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The ˇCech complex ˇC(f) of f is defined to be its csk0-image: ˇC(f) :≡ (y : Y)×csk0((x : X)×(fx = y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 19 We will justify this definition by calculating the type of n-simplices of ˇC(f) when both X and Y are 0-skeletal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f : X → Y be a simplicially crisp map between 0-skeletal types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then ˇC(f)n ≃ X ×Y ···×Y X ≃ (y : Y)×((x : X)×(fx = y))n+1 is the (n+1)-fold pullback of f along itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We calculate: ˇC(f)n :≡ sk0(∆[n] → ˇC(f)) ≡ sk0(∆[n] → (y : Y)×csk0((x : X)×(fx = y))) ≃ sk0((σ : ∆[n] → Y)×((i : ∆[n]) → csk0((x : X)×(fx = σi)))) Since Y is 0-skeletal, any map ∆[n] → Y is constant, so we may continue: ≃ sk0((y : Y)×(∆[n] → csk0((x : X)×(fx = y)))) Since, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4, sk0 ∆[n] = [n], we may use the adjointness of sk0 and csk0 to continue: ≃ sk0((y : Y)×csk0([n] → (x : X)×(fx = y))) Now, we may use Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='8 of [47] to pass the sk0 into the pair type, and then use that sk0 csk0 = sk0 to continue: ≃ ((u : sk0Y)×let u := ysk0 insk0([n] → (x : X)×(fx = y))) However, all types involved are already 0-skeletal, so we may remove the sk0s: ≃ ((y : Y)×([n] → (x : X)×(fx = y))) ≃ (y : Y)×((x : X)×(fx = y))n+1 This last type is the (n+1)-fold pullback of f along itself, displayed in terms of its diagonal map to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We can prove modally that the realization of the ˇCech nerve of a map f : X → Y is the image im f of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This follows from Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 of Real Cohesion [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If A is 0-coskeletal, then reA is a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As a corollary, re(csk0 X) ≃ ∥X∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In [47], this theorem is said to rely on the crisp Law of Excluded Middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' However a glance at the proof reveals that this assumption is only used to assume the decidable equality of sk0 ∆[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since we know that sk0 ∆[1] ≃ {0,1} has decidable equality, the proof goes through without assuming crisp LEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For a map f : X → Y, the realization re ˇC(f) of the ˇCech nerve is the realization reim f of the image of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If furthermore Y is 0-skeletal, then re ˇC(f) ≃ im f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We compute: re ˇC(f) ≡ re((y : Y)×csk0 fib f (y)) ≃ re((y : Y)×recsk0 fib f (y)) ≃ re((y : Y)× ��fib f (y) ��) ≡ reim f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Now if Y is 0-skeletal, then by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='17 of [47], im f is also 0-skeletal, since it is a subtype of a 0-skeletal type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, reim f ≃ im f, so that in total re ˇC(f) ≃ im f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 20 As an application of ˇCech nerves, we can see how to extract coherence data for higher groups from their deloopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If we take the ˇCech nerve of the inclusion ptBG : ∗ → BG of the base point of the delooping of G, we recover a simplicial type whose simplicial identities give coherences for the multiplication of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let G be a crisp, 0-skeletal higher group — a 0-skeletal type identified with the loops of a pointed, 0-connected type BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then ˇC(ptBG)n ≃ Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Furthermore, d1 : ˇC(ptBG)2 → ˇC(ptBG)1 is the product of the projections d0 and d2 : G2 → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13, we know that ˇC(ptBG)n ≃ (e : BG)×((x : ∗)×(ptBG∗ = e))n+1 ≃ (e : BG)×(ptBG = e)n+1 ≃ (e : BG)×(ptBG = e)×(ptBG = e)n ≃ (ptBG = ptBG)n = Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the second to last step, we contract (e : BG)×(ptBG = e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Now, di : ˇC(ptBG)2 → ˇC(ptBG)1 is given by forgetting the ith component of the list (e,(a,b,c)) : (e : BG) × (ptBG = e)n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, d0(e,(a,b,c)) = (e,(b,c)) d1(e,(a,b,c)) = (e,(a,c)) d2(e,(a,b,c)) = (e,(a,b)) Contracting away e and the first element of the pair, we get the three equations d0(ba−1,ca−1) = cb−1 d1(ba−1,ca−1) = ca−1 d2(ba−1,ca−1) = ba−1 and indeed, we have d1(ba−1,ca−1) = d0(ba−1,ca−1)d2(ba−1,ca−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is equivalent, but not quite the same, as the standard presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It amounts to d0(g,h) = hg−1 d1(g,h) = h d2(g,h) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Using the ˇCech nerve, we can extract all the coherence conditions governing a homomorphism of higher groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We first note that the realization of the ˇCech nerve of a group is a delooping of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let G be a 0-skeletal higher group with simplicially crisp delooping BG, and let ˇC(G) be the ˇCech nerve of the basepoint inclusion ptBG : ∗ → BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then the projection fst : ˇC(G) → BG is a re-unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9 of [34], BG is 0-skeletal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Axiom ∆sk0, it is therefore re-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, to show that fst : ˇC(G) → BG is a re-unit, it suffices to show that it is re-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since BG is 0-connected, it suffices to show that the fiber over the base point is re-connected, and this fiber is equivalent to csk0 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This follows by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' recsk0 G is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 21 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let G and H be 0-skeletal higher groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then the type of homomorphisms G → H is equivalent to the type of pointed maps ˇC(G) ·→ ˇC(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Recall that a homomorphism of higher groups is by definition a pointed map between their deloopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' That is, a homomorphism ϕ : G → H is equivalently a diagram as on the left, while a pointed map between the ˇCech nerves is a diagram as on the right: � � � � � � � � � � � ∗ ∗ BG BH ptBG Bϕ ptBH � � � � � � � � � � � ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='≃ � � � � � � � � � � � ∗ ∗ ˇC(G) ˇC(H) ptBG f ptBH � � � � � � � � � � � We are aiming for an equivalence between these two types, which we may present as a one-to-one correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' So,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' to Bϕ : BG → BH and ptBG : ptBH = Bϕ(ptBG) and f : ˇC(G) → ˇC(H) and ptf : pt ˇC(H) = f(pt ˇC(G)) associate the type (□ : (x : ˇC(G)) → (Bϕ(fstx) = fst(fx)))×(ptBϕ ·□(pt ˇC(G)) = fst∗ ptf ) which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' diagrammatically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' is the type of witnesses that the following diagram commutes: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(G) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(H) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='Bϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='fst ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='fst ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='To show that this gives a one-to-one correspondence means showing that the types of diagrams ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(G) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(H) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='Bϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(G) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(H) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='Bϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='are both contractible,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' the left for any homomorphism ϕ and the right for any pointed map f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let ϕ be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since by definition ˇC(G) and ˇC(H) were the csk0-factorizations of the basepoint inclusions, there is a unique filler of this square: ∗ ∗ ˇC(H) ˇC(G) BG BH Bϕ ptBG ptBH ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But this is precisely a rearrangement of the diagram on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Similarly, if f is a pointed map, then re f : BG → BH makes the diagram on the right commute, and by the universal property of the re-unit this is the unique such map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 Global Equivariant Cohesion In Global Homotopy Theory and Cohesion [41], Rezk shows that the ∞-topos of global equivariant homotopy types is cohesive over the ∞-topos of homotopy types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' While Rezk constructs his site out of all compact Lie groups, we will follow Sati and Schreiber [43] in restricting our attention to the finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The global orbit category Glo is defined to be the full subcategory of homotopy types spanned by the deloopings BG of finite groups G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is a (2,1)-category, and the global equivariant topos is defined to be the ∞-category of homotopy type valued presheaves on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There is an adjoint quadruple connecting the global equivariant topos and the topos of homotopy types: S Gloop S colim ∆ Γ ∇ colimX is the colimit of the functor X : Gloop → S , which takes the strict quotient of the global equivariant homotopy type X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ∆S is the inclusion of constant functors: ∆X(BG) :≡ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will refer to such equivariant types as invariant types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ΓX :≡ X(∗) is the evaluation at the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is known as the homotopy quotient of the global equivariant homotopy type X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ∇S is the Yoneda embedding: ∇S(BG) :≡ S (BG,S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This adjoint quadruple gives rise to the cohesive modalities < ⊣ ⊂ ⊣ ≺ of equivariant cohesion: The (“equivariant shape”) strict quotient modality X �→ < X sends a global equivariant type to its strict quotient, considered as an invariant type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The (“equivariant flat”) homotopy quotient modality X �→ ⊂ X sends a global equivariant type to its homotopy quotient, considered as an invariant type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Internally speaking, we say that an equivariantly crisp type is invariant when it is ⊂ modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The (“equivariant sharp”) orbisingular modality X �→ ≺ X sends a global equivariant type to its homotopy quotient, but considered with its natural equivariance via maps from the deloopings of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our axioms for global equivariant cohesion are quite straightforward: Axiom 4 (Global Equivariant Axioms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The type family ≺ B : FinGrp → Type sending a finite group G to ≺ BG detects equivariant continuity and connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The types ≺ BG for finite groups G are the orbi-singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the definition above, we may recover X(BG) (considered with its natural equivariance) as ≺ ( ≺ BG → X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The family ≺ BG for finite groups G is a large family, but we may reduce it to a small family by noting that the type of finite groups is essentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is a useful observation, since it allows us to conclude that < , defined by nullifying all ≺ BG, is an accessible modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Global equivariant cohesion shares a feature with Shulman’s continuous real cohesion: both are definable in the sense that the types which detect continuity and connectivity are definable without axioms in the type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is not the case for simplicial cohesion, which appears to require postulating the 1-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is not clear to us whether there are any general features shared by definable cohesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 23 In Proper Orbifold Cohomology [43], Sati and Schreiber work with equivariant differential cohesion to give an abstract account of the differential cohomology of orbifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We can prove some of their lemmas easily in global equivariant cohesion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' we will return to prove the lemmas relating equivariant and differential cohesion in the upcoming §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The following lemma appears as Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='62 in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We have the following equivalences for the generic orbi-singularities ≺ BG: <≺ BG ≃ ∗ ⊂≺ BG ≃ BG ≺≺ BG ≃ ≺ BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The first equivalence follows by the assumption that ≺ BG detects equivariant connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The second follows by combining Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='22 of [47] to see that ⊂< BG ≃ ⊂ BG with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9 of [34] to note that since G is crisply ⊂ modal (as a finite set), BG is as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The third is simply the idempotence of ≺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The following lemma is a slight strengthening of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='65 of [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let X be an equivariantly crisp set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then X is both invariant ( ⊂ modal) and orbi-singular ( ≺ modal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In both cases we will use that ≺ BG detects equivariant continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To show that X is invariant, we must show that the ⊂ counit is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6, it suffices to show that the map ⊂ ( ≺ BG → ⊂ X) → ⊂ ( ≺ BG → X) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But ≺ is a lex modality and BG is 0-connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' therefore, ≺ BG is 0-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Furthermore, ⊂ X is a set since X is, using Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7 of [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, the above map is equivalent to ⊂ ε : ⊂⊂ X → ⊂ X, which is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Similarly, to show that X is orbi-singular, it suffices to show that the map ⊂ ( ≺ BG → X) → ⊂ (BG → X) given by pre-composing with the ⊂ counit of ≺ BG is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But again, by the connectivity of ≺ BG and BG, this map is equivalent to the identity ⊂ X → ⊂ X, which is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Miller’s theorem (formerly the Sullivan conjecture) states that the space of maps BG → X with G a finite group and X a finite cell complex is equivalent to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In equivariant modal terms, this says that finite cell complexes (the closure of the class {/0, ∗} under pushout) are ≺ modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is not likely that this theorem could be proven on purely modal grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' However, if Miller’s theorem were proven in ordinary HoTT, then the modal statement could be proven in a manner similar to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4 but instead of appealing to the truncatedness of X, appealing to the proof of Miller’s theorem (since any crisp finite cell complex is ⊂ modal as a crisp pushout of ⊂ types).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4 Topological Toposes Johnstone defined his topological topos in [25] in order to provide a topos of spaces for which the geometric realization of simplicial sets was a geometric morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The problem with using a real-cohesive topos for this purpose (as suggested previously by Lawvere) is the failure of the analytic lesser limited principle of omniscience which says that RD = (−∞,0]∪[0,∞) (see Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7 of [47] and the discussion in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 of ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This failure means that gluing together simplices along their (closed) faces gives the wrong topology on the resulting space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Johnstone remedies this by changing the test space from the real numbers to the walking convergent sequence N∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The walking convergent sequence may be defined internally as the set of monotone functions N → {0,1} (see for example [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, it is rather straightforward to give an internal axiomatization for Johnstone’s topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 24 Axiom 5 (Topological Focus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The topological focus is determined by asserting that N∞ detects topological conti- nuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may also be able to determine condensed homotopy types as in [19] — or rather the similar but more topos-theoretic pyknotic homotopy types of [10] — using a similar axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Define a profinite set to be the limit of a crisp diagram of finite sets indexed by a discrete (♭-modal) partially ordered set with decidable order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may then assert that the family of profinite sets detects condensed continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' However, we do not know if these axioms are sufficient for proving theorems in these topological toposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 5 Multiple Focuses Now, we turn our attention to generalities on possible relationships between different focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For this section, fix two focuses ♥ and ♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First, we should show that the focuses do indeed commute: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For any type B, the map ♯♥♯♣B → ♯♣♯♥B defined by x �→ x♯♥♯♣ ♯♥♯♣ is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Furthermore, the maps ♯♥♯♣B → ♯♥♣B and ♯♣♯♥B → ♯♥♣B defined by x �→ x♯♥♯♣ ♯♥♣ x �→ x♯♣♯♥ ♯♥♣ are also equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The first map is well-defined because the use of ♯♥- and ♯♣-introduction means that the assumption x becomes crisp for both ♥ and ♣, so we may apply ♯♥- and then ♯♣-elimination to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may similarly define an inverse by x �→ x♯♣♯♥ ♯♣♯♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These maps are definitional inverses by the computation rules for ♯s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The other maps are similarly well defined since being crisp for both ♥ and ♣ means being crisp for focus ♥♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The inverse may be defined in the straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We also note that the ordering of focuses is reflected in the containment of their ♯-modal (and so also ♭-modal) types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♣ ≤ ♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then any ♯♣-modal type is ♯♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since ♣ ≤ ♥ is defined to mean ♥♣ ≡ ♣, we know that ♯♥♣A ≡ ♯♣A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By assumption ♯♣A ≃ A, and chaining this with the commutativity equivalence of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 ♯♥A ≃ ♯♥♯♣A ≃ ♯♥♣A ≡ ♯♣A ≃ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Tracing these simple equivalences through, this does indeed give an inverse to the ♯♥-unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We can similarly show that the ♭s commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is made simpler through the use of crisp induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let A be an ♥♣-crisp type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then ♭♥♭♣A → ♭♣♭♥A defined by u �→ let v♭♥ := uin(let w♭♣ := vin(w♭♥)♭♣) is an equivalence, natural in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In words, the map is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Performing ♭♥-induction on u : ♭♥♭♣A gives an assumption v :♥ ♭♣A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A second induction on the term v : ♭♣A then gives us an assumption w :♥♣ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This second induction is ‘♥-crisp ♭♣-induction’: the resulting assumption w inherits the ♥-crispness of term v and gains ♣-crispness from the removal of ♭♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we form (w♭♥)♭♣ by applying ♭-introduction twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A map in the other direction is constructed in the same way, and then the proofs these are inverse are immediate by another pair of inductions each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We note also that the ♭-inductions commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭♥-induction and ♭♣-induction commute: let u♭♣ := (let v♭♥ := M inN)inC = let v♭♥ := M in(let u♭♣ := N inC) (when this is well-typed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=', v does not occur in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First using uniqueness of ♭♣, we have let u♭♣ := (let v♭♥ := M inN)inC = let w♭♥ := M in(let u♭♣ := (let v♭♥ := w♭♥ inN)inC) ≡ let w♭♥ := M in(let u♭♣ := N[w/v]inC) ≡ let v♭♥ := M in(let u♭♣ := N inC) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 Commuting Cohesions Now let’s turn our attention to the relationships between two commuting cohesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♥ and ♣ are both cohesive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If a ♥♣-crisp type A is ♭♥-modal, then S♣A is still ♭♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We need to produce an inverse to the counit ε♥ : ♭♥S♣A → S♣A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First construct the composite A s−→ ♭♥A ♭♥η♣ −−−→ ♭♥S♣A where A s−→ ♭♥A is the assumed inverse to ε♥ : ♭♥A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The type ♭♥S♣A is certainly ♭♣-modal by commutativity of ♭♥ and ♭♣: ♭♣♭♥S♣A ≃ ♭♥♭♣S♣A ≃ ♭♥S♣A and therefore the above map factors through i : S♣A → ♭♥S♣A, our purported inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For one direction, consider the naturality square for the counit ε♥: ♭♥S♣A ♭♥♭♥S♣A S♣A ♭♥S♣A ε♥ ♭♥i ε♥ i The map ♭♥i is equal to the comultiplication δ♥ : ♭♥A → ♭♥♭♥A defined by a♭♥ �→ a♭♥♭♥, because both are inverse to the map ♭♥ε♥ : ♭♥♭♥A → ♭♥A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' And so the bottom composite in the square is equal to ε♥ ◦ δ♥, which is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 26 In the other direction, because S♣A is S♣-modal, it suffices to show that the composite A → S♣A → ♭♥S♣A → S♣A is equal to the unit η♣ : A → S♣A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For this we have the following commutative diagram: A ♭♥A A S♣A ♭♥S♣A S♣A ∼ η♣ ♭♥η♣ ε♥ η♣ i ε♥ The left square commutes by the definition of i, the right square by naturality of ε♥, and the composite along the top is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♥ and ♣ are both cohesive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then S♥S♣A → S♣S♥A is an equivalence for any ♥♣-crisp type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The map η♣ ◦η♥ : A → S♥A → S♣S♥A factors through S♥S♣A, because S♣S♥A is ♭♣-modal, as a type of the form ♭♣X, and also ♭♥-modal, by the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The map the other way is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To show these are inverses it suffices to show that they become so after precomposition with the composites of the units, because S♣S♥A and S♥S♣A are ♥♣-discrete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' this is immediate by the definition of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We might also hope that, say, ♭♥ and S♣ commute in general, but there is a useful sanity check that shows this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the bare type theory with no axioms, there is nothing that prevents interpretation in a model where ♭♥ ≡ ♭♣ and S♥ ≡ S♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In ordinary cohesive type theory it is certainly not the case that ♭ and S commute, and so ♭♥S♣ ≃ S♣♭♥ cannot be provable without further assumptions on ♥ and ♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A sufficient assumption on our focuses to make ♭♥ and S♣ commute in this way is the following: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that G :♥♣ I → Type and H :♥♣ J → Type detect ♥ and ♣ connectivity respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We say that focuses ♥ and ♣ are orthogonal if Gi is ♭♣-modal for all i, and Hj is ♭♥-modal for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our present goal is to show that this indeed makes ♭♥S♣ ≃ S♣♭♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will in fact only use that the Gi are ♭♣-modal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' of course the dual results, flipping ♥ and ♣, require the other half of orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let ♥ and ♣ be cohesive focuses that are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then for any ♣-crisp A, if A is S♥-modal, ♭♣A is still S♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our goal is to show that ♭♣A is equivalent to Gi → ♭♣A via precomposition by Gi → 1, for any i : I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We easily check that the type Gi → ♭♣A is ♣-discrete: for any Hj, Hj → (Gi → ♭♣A) ≃ Hj → (Gi → ♭♣A) ≃ Gi → (Hj → ♭♣A) ≃ Gi → ♭♣A because ♭♣A is ♣-discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then, by adjointness of S♣ and ♭♣: (Gi → ♭♣A) ≃ ♭♣(Gi → ♭♣A) ≃ ♭♣(S♣Gi → A) ≃ ♭♣(Gi → A) ≃ ♭♣A 27 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5 (Crisp S♥-induction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♥ and ♣ are cohesive and orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If B is ♥-discrete and ♣-crisp, then for any ♣-crisp A the map ♭♣(♭♣S♥A → B) → ♭♣(♭♣A → B) given by precomposition by ♭♣η♥ : ♭♣A → ♭♣S♥A is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As a rule, crisp S-induction would be written: ♣\\Γ,x :♣ S♥A ⊢ C ♣\\Γ,x :♣ S♥A ⊢ w : is-♭♥-modal(C) ♣\\Γ ⊢ M : S♥A ♣\\Γ,u :♣ A ⊢ N : C[uS♥/x] Γ ⊢ (let uS♥ := M inN) : C[M/x] Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We deploy the usual trick for deriving crisp induction principles: using ♯♣ to move the ♭♣ out of the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Crucially, the previous proposition is what allows us to apply the universal property of S♥ on maps into ♯♣B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭♣(♭♣S♥A → B) ≃ ♭♣(S♥A → ♯♣B) ≃ ♭♣(A → ♯♣B) ≃ ♭♣(♭♣A → B) Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If ♥ and ♣ are orthogonal and cohesive focuses, then S♥ and ♭♣ commute on ♣-crisp types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is now straightforward induction on S♥ and ♭♣ in both directions, using crisp S♥-induction when defining ♭♣S♥A → S♥♭♣A and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4 when defining S♥♭♣A → ♭♣S♥A to know that ♭♣S♥A is ♥-discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If ♥ and ♣ are orthogonal and cohesive focuses, then ♭♥ and ♯♣ commute on ♥♣-crisp types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On ♥♣-crisp types, ♭♥♯♣ is right adjoint to ♭♣S♥, and ♯♣♭♥ is right adjoint to S♥♭♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, if ♭♣S♥X ≃ S♥♭♣ for a ♥♣-crisp type X, then also ♭♥♯♣X ≃ ♯♣♭♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Next we investigate a relationship between S and ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Again, this depends on a relationship between the families which detect continuity and connectivity of the two focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♣ is cohesive, and that the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' H :♥♣ I → Type detects ♣-connectivity, and I is ♭♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Hi is ♭♥-modal for all i :♥ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' (In particular, if ♥ and ♣ are orthogonal) Then if X is S♣-modal then ♯♥X is also S♣-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since H detects ♣-connectivity, it suffices to show that ♯♥S♣X is Hi-null for every i : I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since I is ♭♥-modal, we may assume that i :♥ I is ♥-crisp by ♭♥-induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then we can compute: (Hi → ♯♥X) ≃ ♯♥(Hi → ♯♥X) ≃ ♯♥(♭♥Hi → X) ≃ ♯♥(Hi → X) since Hi was assumed ♭♥-modal ≃ ♯♥X since X is S♣-modal Tracing upwards through this series of equivalences shows that the composite is indeed the inclusion of constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 28 On the opposite extreme of orthogonality, we can see that if the Gi which detect the connectivity of ♥ are S♣-connected, then any S♣-modal type is S♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♥ and ♣ are cohesive focuses where G : I → Type detects the connectivity of ♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then the following are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Every Gi is S♣-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Any S♣-modal type is S♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that Gi is S♣-connected for all i and that X is S♣-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may compute: (1 → X) ≃ (S♣Gi → X) ≃ (Gi → X) In the first equivalence, we use that Gi is S♣-connected, and in the second that X is S♣-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We conclude that X is S♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Conversely, suppose that any S♣-modal type is S♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then in particular S♣Gi is S♥-modal, so that the identity map S♣Gi → S♣Gi factors through S♥S♣Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 we have S♥S♣Gi ≃ S♣S♥Gi ≃ S♣∗ ≃ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, the identity of S♣Gi factors through the point, which means it is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6 Examples with Multiple Focuses In this section, we will see examples with multiple focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In particular, we will see simplicial real cohesion, equivariant differential cohesion, and supergeometric cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 Simplicial Real Cohesion We assume two basic focuses: the real (continuous or differential) focus S ⊣ ♭ ⊣ ♯, and the simplicial focus re ⊣ sk0 ⊣ csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will write R for whichever flavor of real numbers is used in the real cohesive focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will assume both the axioms of real cohesion and simplicial cohesion, as well as the following axiom relating the two focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Axiom 6 (Simplicial Real Cohesion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume that the real focus and simplicial focus are orthogonal — which is to say, R is 0-skeletal and that ∆[1] is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Furthermore, we assume that S is computed pointwise: for any simplicially crisp type X, the action (ηS)n : Xn → (SX)n of the S-unit of X on n-simplices is itself a S-unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our goal in this section will be to prove that if M is a 0-skeletal type — to be thought of as a “manifold”, having only real-cohesive structure but no simplicial structure — and U is a good cover of M — one for which the finite intersections are S-connected whenever they are inhabited — then the homotopy type SM of M may be constructed as the realization of a discrete simplicial set — namely, the ˇCech nerve of the open cover, with each open replaced by the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let M be a 0-skeletal type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A cover of M consists of a discrete 0-skeletal index set I, and a family U : I → (M → Prop) of subobjects of M so that for every m : M there is merely an i : I with m ∈ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may assemble a cover into a single surjective map c : � i:IUi → M, where � i:I Ui :≡ (i : I)×(m : M)×(m ∈ Ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A cover U : I → (M → Prop) is a good cover if for any n : N and any k : [n] → I, the S-shape of the intersection � i:[n] Uk(i) :≡ (m : M)×((i : [n]) → (m ∈ Uk(i))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' is a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' That is, S(Uk(0) ∩···∩Uk(n)) is contractible whenever there is an element in the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 29 We begin with a few ground-setting lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let U : I → (M → Prop) be a simplicially crisp cover, and let c : � i:IUi → M be the associated covering map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Consider the projection π : ˇC(c) → csk0 I defined by (m,z) �→ (fstzcsk0)csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Over a simplicially crisp n-simplex k : ∆[n] → csk0 I, we have fibπn(ksk0) ≃ � i:[n] Uk(isk0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As a corollary, we have that ˇC(c)n ≃ (k : I[n])× � i:[n] Uk(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We compute: fibπn(ksk0) ≡ (x : ˇC(c)n)×(πnx = ksk0) ≃ ((m,z) : (m : M)×((i : I)×(m ∈ Ui))[n])×(πne(m,z) = ksk0) ≃ (m : M)×(K : [n] → I)×(p : (i : [n]) → (m ∈ UK(i)))×(πne(m,i �→ (K(i), p)) = ksk0) Here, e(m,z) is image under the equivalence from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' When all the modal dust settles, we will be left knowing that πne(m,i �→ (K(i), p)) : sk0(∆[n] → csk0 I) is the unique correspondent to K : [n] → I under the sk0 ⊣ csk0 adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, we may contract K away with ksk0 in the above type to get: ≃ (m : M)×((i : [n]) → m ∈ Uk(isk0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For the next lemma, we will need to know that S commutes with csk0 on suitably crisp types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For any simplicially crisp type X, we have that Scsk0 X ≃ csk0 SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We know by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9 that csk0 SX is S-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We therefore have a map Scsk0 X → csk0 SX given as the unique factor of csk0(−)S : csk0 X → csk0 SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will show that this map is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since it is crisp, it suffices to show that it is an equivalence on n-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To that end, we compute: sk0(∆[n] → Scsk0 X) ≃ Ssk0(∆[n] → csk0 X) ≃ Ssk0([n] → X) ≃ sk0(S([n] → X)) ≃ sk0([n] → SX) ≃ sk0(∆[n] → csk0 SX) It remains to show that this is indeed the right equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since the first equivalence in the series above is given as the inverse of (−)S n : (csk0 X)n → (Scsk0 X)n, it suffices to check that given a crisp z : ∆[n] → csk0 X, (zsk0)S corresponds under the above equivalences to (csk0(−)S ◦z)sk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First, we send (zsk0)S to ((z◦(−)sk0)sk0)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then, we send it to ((z◦(−)sk0)S)sk0, and then to (i �→ (z(isk0)S))sk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we map this to (i �→ ((z(isk0sk0)S))csk0 sk0, which does equal (csk0(−)S ◦z)(i) ≡ (z(i)S)csk0 at i : ∆[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let U : I → (M → Prop) be a simplicially crisp cover, and let c : � i:IUi → M be the covering map itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Consider the “projection” π : ˇC(c) → csk0 I defined by (m,z) �→ (fstzcsk0)csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then U is a good cover if and only if the restriction π : ˇC(c) → imπ is a S-unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 30 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since csk0 and S commute by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 and I is discrete, csk0 I is also discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As the subtype of a discrete type, imπ is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, it suffices to show that π : ˇC(c) → imπ induces an equivalence S ˇC(c) ∼−→ imπ if and only if the cover U is good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since π is crisp, S ˇC(c) → imπ is an equivalence if and only if it is an equivalence on all n-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On n-simplices, this map (at the top of the following diagram) is equivalent to the map on the bottom of the following diagram: (S ˇC(c))n (imπ)n S(ˇC(c)n) imπn S � (k : I[n])× � i:[n]Uk(i) � (k : I[n])×S �� i:[n]Uk(i) � (k : I[n])×∃� i:[n]Uk(i) The bottom map is an equivalence if and only if the cover is good, and so we conclude the same for the top map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we can piece these lemmas together for our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let U : I → (M → Prop) be a simplicially crisp good cover of a 0-skeletal type M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let π : ˇC(U) → csk0 I be the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then reimπ ≃ SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This exhibits the shape SM as the realization of a discrete (simplicial) set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since the cover is good, we have that imπ ≃ SˇC(U), so that reimπ ≃ reSˇC(U) ≃ Sre ˇC(U) ≃ SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Now, since I is a set and csk0 is a lex modality, csk0 I is also a set and so imπ is a set as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Furthermore, since subtypes of discrete types are discrete by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='17 of [47], and csk0 I is discrete since by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 S and csk0 commute on simplicially crisp types, imπ is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 Equivariant Differential Cohesion In Proper Orbifold Cohomology, Sati and Schreiber work in equivariant differential cohesion to describe the differ- ential cohomology of orbifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This cohesion involves both the equivariant focus < ⊣ ⊂ ⊣ ≺ and the differential (real-cohesive) focus S ⊣ ♭ ⊣ ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this section, we will assume both the axioms of equivariant cohesion and differ- ential real cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will refer to the smooth reals by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Unlike the simplicial real cohesive case, we do not need to add additional axioms to ensure that the equivariant and differential cohesion are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Equivariant and differential cohesion are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' That is: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The smooth reals R are invariant ( ⊂ modal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For any finite group G, ≺ BG is discrete (♭-modal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since the smooth reals are a set, they are invariant by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Similarly, since BG is discrete and hence S-modal (by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9 of [34]), ≺ BG is still S-modal by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 31 The following lemma appears as Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='67 of [43], and is proven quickly with our general lemmas concern- ing orthogonal cohesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that X is both differentially and equivariantly crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then ⊂ SX ≃ S ⊂ X ⊂ ♭X ≃ ♭ ⊂ ♭X ⊂ ♯X ≃ ♯ ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The first equivalence follows by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The second equivalence follows by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The third follows by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 Supergeometric Cohesion In his habilitation, Differential Cohomology in a Cohesive ∞-Topos [44], Schreiber describes an increasing tower of adjoint modalities which appear in the setting of supergeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The setting for supergeometric cohesion — called “solid cohesion” in ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' — is sheaves on the opposite of a category of super C ∞-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Schreiber calls these sheaves super formal smooth ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Specifically, the site is (the opposite of) the full subcategory of the category of super commutative real algebras spanned by objects of the form C ∞(Rn)⊗W ⊗ΛRq where W is a Weil algebra — a commutative nilpotent extension of R which is finitely generated as an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The factor C ∞(Rn)⊗W is even graded, while the Grassmannian ΛRq is odd graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' See Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13 of [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The inclusion of algebras of the form C ∞(Rn) ⊗W has a left and a right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The left adjoint is given by projecting out the even subalgebra, and the right adjoint is given by quotienting by the ideal generated by the odd graded elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This gives rise to an adjoint quadruple between the resulting toposes of sheaves and thus an adjoint triple of idempotent adjoint (co)monads on the topos of super formal smooth ∞-groupoids: ⇒ ⊣ ⇝ ⊣ Rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Of these, ⇒ and Rh are idempotent monads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' However, ⇒ does not preserve products, and so does not give an internal modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The action of Rh is easy to define: RhX(C ∞(Rn)⊗W ⊗ΛRq) := X(C ∞(Rn)⊗W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' That is, RhX is defined by evaluating at the even part of the superalgebra in the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may characterize it internally by localizing at the odd line R0|1, which is the sheaf represented by the free superalgebra on one odd generator ΛR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We turn to the internal story now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The topos of super formal smooth ∞-groupoids also supports the differential real cohesive modalities S ⊢ ♭ ⊢ ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These destroy all geometric structure — super and otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For this reason, we will work with the lattice {diff < super < ⊤} of focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The modalities of the super focus are ⇝ ⊢ Rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will refer to ⇝ X as the even part of X, while Rh is known as the rheonomic modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume the following axioms for supergeometric or solid cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Axiom 7 (Solid Cohesion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Solid cohesion uses the focus lattice {diff < super < ⊤}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We use the definition of real superalgebras due to Carchedi and Roytenberg [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume a commutative ring R1|0 satisfying the axioms of synthetic differential geometry (as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 of [36]) known as the smooth reals or the even line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume an R1|0-module R0|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There is furthermore a bilinear multiplication R0|1×R0|1 → R1|0 which sat- isfies a2 = 0 for all a : R0|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Together these axioms imply that R1|1 := R1|0×R0|1 is a R1|0-supercommutative superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume the following odd form of the Kock-Lawvere axiom: For any function f : R0|1 → R0|1 with f(0) = 0, there is a unique r : R1|0 with f(x) = rx for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume that R0|1 is ⇝-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume that R1|0 detects differential connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume that a type is Rh-modal if and only if it is R0|1-null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It might seem prudent to instead ask that differential connectivity is detected by the family con- sisting of both R1|0 and R0|1, since we want S to nullify all representables Rn|q, but it suffices to test with R1|0 since R0|1 admits an explicit contraction by its R1|0-module structure (appealing to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='10 of [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2, any ♯-modal type is Rh-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But also every ♭-modal type is Rh-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If X is S-modal (and, in particular, if X is ♭-modal), then X is Rh-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since R0|1 is S-connected due to its explicit contraction by the scaling of its module structure, any S-modal type is R0|1-null, and therefore Rh-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' References [1] LICS ’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Oxford, United Kingdom: Association for Computing Machinery, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ISBN: 978-1-4503-5583- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [2] Andreas Abel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “A Polymorphic Lambda-Calculus with Sized Higher-Order Types”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' June 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: http://www2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='de/~abel/diss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [3] Andreas Abel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Polarised subtyping for sized types”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Mathematical Structures in Computer Science 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5 (2008), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 797–822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1017/S0960129508006853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [4] Benedikt Ahrens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The Univalence Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' arXiv: 2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='06275 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='CT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [5] Thorsten Altenkirch, Paolo Capriotti, and Nicolai Kraus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Extending homotopy type theory with strict equal- ity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Computer Science Logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Leibniz Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Schloss Dagstuhl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Leibniz-Zent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=', Wadern, 2016, Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 21, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='CSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [6] Mathieu Anel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Enveloping ∞-topoi”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Seminar on Higher Homotopical Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='youtube.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' by Mathieu Anel and Gabriel Catren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Cambridge University Press, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 155–257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1017/9781108854429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [8] Danil Annenkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Two-Level Type Theory and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' arXiv: 1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='03307 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='LO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [9] Robert Atkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Syntax and Semantics of Quantitative Type Theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' LICS ’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Oxford, United Kingdom: Association for Computing Machinery, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 56–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ISBN: 978-1-4503-5583-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1145/3209108.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1017/S0960129519000197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [12] Guillaume Brunerie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “On the homotopy groups of spheres in homotopy type theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Universit´e de Nice, 2016.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1184/R1/14555691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [17] Felix Cherubini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Formalizing Cartan Geometry in Modal Homotopy Type Theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Karlsruher Institut f¨ur Technologie, July 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Daniel Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Localization in homotopy type theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Higher Structures 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 1–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: https://higher- structures.' metadata={'source': 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10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2178/jsl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7803040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [21] Daniel Gratzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Modalities and Parametric Adjoints”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Comput.' metadata={'source': 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R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Licata and Michael Shulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Adjoint Logic with a 2-Category of Modes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Logical founda- tions of computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 9537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lecture Notes in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Springer, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 219–235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1007/978- 3- 319- 27683- 0_16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: http://dlicata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} 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International Conference on Formal Structures for Computation and Deduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' by Dale Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' FSCD ’17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2017.' 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extensionality, and proof irrelevance in modal type theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Proceedings 16th Annual IEEE Symposium on Logic in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2001, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 221–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1109/LICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='932499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [40] Jason Reed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “A Judgmental Deconstruction of Modal Logic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='cs.' metadata={'source': 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+page_content=' URL: https://faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' edu/~rezk/global-cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [42] Emily Riehl and Michael Shulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “A type theory for synthetic ∞-categories”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: High.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 147–224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1007/s42001-017-0005-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [43] Hisham Sati and Urs Schreiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proper Orbifold Cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: https://ncatlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='org/ schreiber/files/orbi220627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [44] Urs Schreiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Differential cohomology in a cohesive infinity-topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: https://ncatlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='org/ schreiber/files/dcct170811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [45] Urs Schreiber and Michael Shulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Quantum Gauge Field Theory in Cohesive Homotopy Type Theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Workshop on Quantum Physics and Logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4204/EPTCS.' metadata={'source': 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+page_content='07004 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='AT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [47] Michael Shulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Brouwer’s fixed-point theorem in real-cohesive homotopy type theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Mathe- matical Structures in Computer Science 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 856–941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ISSN: 0960-1295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 1017 / S0960129517000147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [48] Jonathan Weinberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Internal sums for synthetic fibered (∞,1)-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2022.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [49] Jonathan Weinberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Two-sided cartesian fibrations of synthetic (∞,1)-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='48550/ ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='00938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='org/abs/2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='00938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [50] Gavin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Wraith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Using the generic interval”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4 (1993), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 259–266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='numdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='org/item/CTGDC_1993__34_4_259_0/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 35 A Proof Sketches for Admissible Rules We sketch proofs that the operations on syntax are admissible, demonstrating the interesting cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' those that involve division (CTX-EXT, ♭-FORM/INTRO/ELIM, ♯-ELIM) or promotion (♯-FORM/INTRO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The ♥Γ and ♥\\Γ context operations extend in the obvious way to telescopes Γ′, so that ♥(Γ,Γ′) ≡ (♥Γ),(♥Γ′) ♥\\(Γ,Γ′) ≡ (♥\\Γ),(♥\\Γ′) Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 (Weakening).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Single-variable weakening is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' WK Γ,Γ′ ⊢J Γ,w :♣ W,Γ′ ⊢J −−−−−−−−−−− Moreover, weakening does not change the size of the derivation tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Induction onJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-FORM ♥\\(Γ,w :♣ W,Γ′) ⊢ A type Γ,w :♣ W,Γ′ ⊢ ♭♥A type There are two subcases: If ♣ ≤ ♥: then ♥\\(Γ,w :♣ W,Γ′) ≡ (♥\\Γ),w :♣ W,(♥\\Γ′), in which case we induct on the type A and reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If ♣ ̸≤ ♥: then ♥\\(Γ,w :♣ W,Γ′) ≡ (♥\\Γ),(♥\\Γ′) ≡ ♥\\(Γ,Γ′), and so already ♥\\(Γ,w :♣ W,Γ′) ⊢ A type and we can reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (promotion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-FORM ♥(Γ,w :♣ W,Γ′) ⊢ A type Γ,w :♣ W,Γ′ ⊢ ♯♥A type By definition ♥(Γ,w :♣ W,Γ′) ≡ ♥Γ,w :♥♣ W,♥Γ′, and so we induct on A (now weakening with a variable of focus ♥♣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For any two focuses ♥ and ♣, the context ♣\\(♥Γ) is an iterated weakening of ♥(♣\\Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This can be checked variablewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Given a variable x :♠ A in Γ, if it survives to ♥(♣\\Γ) as x :♥♠ A, then we must have ♠ ≤ ♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Multiplying by ♥, it follows that ♥♠ ≤ ♥♣ ≤ ♣, and so x :♥♠ A also occurs in ♣\\(♥Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The following equations involving contexts and telescopes hold: (♥Γ′)[s/z] ≡ ♥(Γ′[s/z]) (♥\\Γ′)[s/z] ≡ ♥\\(Γ′[s/z]) 36 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5 (Substitution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' SUBST ♣\\Γ ⊢ s : S Γ,z :♣ S,Γ′ ⊢J Γ,Γ′[s/z] ⊢J [s/z] −−−−−−−−−−−−−−−−−−−− Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Induction onJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Three subcases as usual, for x ∈ Γ, x ≡ z and x ∈ Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The interesting one is x ≡ z: VAR Γ,z :♣ S,Γ′ ⊢ s : S Applying DIVIDE-WK to ♣\\Γ ⊢ s : S gives Γ ⊢ s : S, which can be further weakened to Γ,Γ′ ⊢ s : S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-FORM ♥\\(Γ,z :♣ S,Γ′) ⊢ A type Γ,s :♣ S,Γ′ ⊢ ♭♥A type There are two subcases: If ♣ ≤ ♥: then ♥ \\ (Γ,z :♣ S,Γ′) ≡ (♥ \\ Γ),z :♣ S,(♥ \\ Γ′), in which case we induct, getting (♥ \\ Γ),(♥\\Γ′)[s/z] ⊢ A[s/z] type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This context is equal to ♥\\(Γ,Γ′[s/z]) ctx, so we can reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If ♣ ̸≤ ♥: then ♥ \\ (Γ,z :♣ S,Γ′) ≡ (♥ \\ Γ),(♥ \\ Γ′) ≡ ♥ \\ (Γ,Γ′), and so z does not occur in A, and A ≡ A[s/z], in which case we may reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (promotion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-FORM ♥(Γ,z :♣ S,Γ′) ⊢ A type Γ,z :♣ S,Γ′ ⊢ ♯♥A type By definition ♥(Γ,s :♣ S,Γ′) ≡ ♥Γ,s :♥♣ S,♥Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Applying PROMOTE to ♣\\Γ ⊢ s : S yields ♥(♣\\Γ) ⊢ s : S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3, this can be weakened to ♣(♥\\Γ) ⊢ s : S, whose context is equal to (♥♣)\\(♥Γ), which is of the correct shape to be substituted into ♥Γ,z :♥♣ S,♥Γ′ ⊢ A type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Substitution gives ♥Γ,(♥Γ′)[s/z] ⊢ A[s/z] type, and this context is equal to ♥(Γ,Γ′[s/z]) ctx, so we can reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6 (Promote).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' PROMOTE-CTX Γ ctx ♣Γ ctx −−−− PROMOTE Γ ⊢J ♣Γ ⊢J −−−−− Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First, PROMOTE-CTX is by induction on the length of the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Consider a context Γ,x :♥ A ctx, so that ♥ \\ Γ ⊢ A type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Applying PROMOTE to A gives ♣(♥ \\ Γ) ⊢ A type, which can be weakened to (♣♥) \\ (♣Γ) ⊢ A type by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3, letting us form the context ♣Γ,x :♣♥ A ctx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The variable rule is immediate, because modifying the annotation on a variable does not change whether it is usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 37 Case (division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-FORM ♥\\Γ ⊢ A type Γ ⊢ ♭♥A type Inductively ♣(♥\\Γ) ⊢ A type, which can be weakened to ♥\\(♣Γ) ⊢ A type, and we reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (promotion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-FORM ♥Γ ⊢ A type Γ ⊢ ♯♥A type Inductively ♣(♥Γ) ⊢ A type, and ♣(♥Γ) ≡ ♥(♣Γ), so we may reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7 (Division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DIVIDE-CTX Γ ctx ♣\\Γ ctx −−−−−− DIVIDE-WK Γ ctx ♣\\Γ ⊢J Γ ⊢J −−−−−−−−−−−− Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First DIVIDE-CTX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Consider a context Γ,x :♥ A ctx, so that ♥\\Γ ⊢ A type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There are two cases: If ♥ ≤ ♣: Then ♥\\Γ ≡ (♥♣)\\Γ ≡ ♥\\(♣\\Γ), and so ♥\\(♣\\Γ) ⊢ A type is of the right shape to form ♣\\Γ,x :♥ A ctx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If ♥ ̸≤ ♣: Then ♣\\(Γ,x :♥ A) ≡ ♣\\Γ is well-formed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Now DIVIDE-WK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On terms: Case (variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If x :♥ A is in context ♣\\Γ, then it must also be in Γ and so we may reuse the variable rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-FORM ♥\\(♣\\Γ) ⊢ A type ♣\\Γ ⊢ ♭♥A type We know ♥\\(♣\\Γ) ≡ ♣\\(♥\\Γ), and so inductively ♥\\Γ ⊢ A type, and we can reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (promotion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-FORM ♥(♣\\Γ) ⊢ A type ♣\\Γ ⊢ ♯♥A type ♥(♣ \\ Γ) ⊢ A type may be weakened to ♣ \\ (♥Γ) ⊢ A type (without increasing the size of the derivation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Inductively ♥Γ ⊢ A type, and we can reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 38' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} diff --git a/1dE3T4oBgHgl3EQfnQrW/content/tmp_files/2301.04624v1.pdf.txt b/1dE3T4oBgHgl3EQfnQrW/content/tmp_files/2301.04624v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b9a1cfd8193b633ab7af67179805959873a54eff --- /dev/null +++ b/1dE3T4oBgHgl3EQfnQrW/content/tmp_files/2301.04624v1.pdf.txt @@ -0,0 +1,5794 @@ +Typical Correlation Length of Sequentially Generated Tensor Network States +Daniel Haag,1, 2, 3, ∗ Flavio Baccari,1, 3 and Georgios Styliaris1, 3 +1Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany +2Physik-Department, Technische Universität München, James-Franck-Str. 1, 85748 Garching, Germany +3Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany +(Dated: January 12, 2023) +The complexity of quantum many-body systems is manifested in the vast diversity of their cor- +relations, making it challenging to distinguish the generic from the atypical features. This can be +addressed by analyzing correlations through ensembles of random states, chosen so as to faithfully +embody the relevant physical properties. Here we focus on spins with local interactions, whose corre- +lations are extremely well captured by tensor network states. Adopting an operational perspective, +we define ensembles of random tensor network states in one and two spatial dimensions that admit +a sequential generation. As such, they directly correspond to outputs of quantum circuits with a +sequential architecture and random gates. In one spatial dimension, the ensemble explores the entire +family of matrix product states, while in two spatial dimensions, it corresponds to random isometric +tensor network states. We extract the scaling behavior of the average correlations between two sub- +systems as a function of their distance. Using elementary concentration results, we then deduce the +typical case for measures of correlation such as the von Neumann mutual information and a measure +arising from the Hilbert–Schmidt norm. We find for all considered cases that the typical behav- +ior is an exponential decay (for both one and two spatial dimensions). We observe the consistent +emergence of a correlation length that only depends on the underlying spatial dimension and not +the considered measure. Remarkably, increasing the bond dimension leads to a higher correlation +length in one spatial dimension but has the opposite effect in two spatial dimensions. +I. +INTRODUCTION +The behavior of correlations in quantum many-body +systems is an inherently difficult problem to character- +ize. +Specifying a generic n-particle state requires ex- +ponentially many parameters, a fact which reflects the +enormous variety of correlations possible in the quantum +realm. +Nonetheless, significant insights can be gained +about the nature of correlations by utilizing random en- +sembles of states. A celebrated result along this direction +shows that random states sampled uniformly from the +full Hilbert space of an n-particle system typically ex- +hibit strong correlations, as manifested by a volume law +behavior for the entanglement entropy [1–5]. However, +there is by now clear evidence that the set of physically +relevant states constitutes an exponentially small subset +of the full Hilbert space of an n-particle system [6], bring- +ing into question the relevance and utility of conclusions +obtained under the assumption of uniform sampling from +the full Hilbert space. +For quantum spin systems with local interactions, +tensor network states have been exceedingly successful +at capturing relevant properties [7]. +They exhibit an +area law for the entanglement entropy by construction, +and are, therefore, good candidates to represent many +physically relevant many-body states. Their preeminent +one-dimensional representatives, matrix product states +(MPS), have been shown to represent faithfully ground +states of gapped local Hamiltonians [8–10] and have given +∗ daniel.haag@tum.de +rise to the complete classification of topological phases of +matter in one dimension [11, 12]. MPS have been also +generalized to their counterparts in two (or more) spa- +tial dimensions, projected entangled pair states (PEPS). +While only a weaker link between local Hamiltonians +and PEPS has been proven rigorously, two-dimensional +PEPS are known to efficiently represent a wide class of +strongly correlated states [7, 13], including states with +power-law [14] and topological correlations [15–17]. +The importance of defining ensembles of random tensor +network states for the purpose of exploring typical prop- +erties of physically relevant states has been recognized +more than a decade ago [18]. MPS ensembles have been +utilized to gain insights into, among other things, the typ- +icality of expectation values of local observables [18, 19], +equilibration under Hamiltonian time evolution [20], the +entropy of subsystems [21], non-stabilizerness [22], and, +most relevant for this work, the behavior of correla- +tions [23–27]. In particular, correlation functions of ran- +dom inhomogeneous MPS (that is, MPS whose local +tensors can be different) were shown to exhibit almost +surely an exponential decay [24, 28]. +A qualitatively +similar behavior was observed also for correlation func- +tions of translation-invariant MPS and PEPS with ran- +dom Gaussian entries [25]. Instead of incorporating the +randomness directly at the level of states, one can also +consider random local Hamiltonians and examine their +ground states. The typical behavior of correlations for +this case was found to depend on the nature of random- +ness, allowing for both long- and short-range correlated +states [23, 29, 30]. +Here, we approach the problem of typical correlations +arXiv:2301.04624v1 [quant-ph] 11 Jan 2023 + +2 +FIG. 1. Sequential generation of MPS and isoTNS. Each circle represents a site of the finalized state and boxes represent the +isometries of the sequential generation. (1a) Sequential generation of an MPS with physical dimension d and bond dimension +D. The diamond indicates the origin of the process. (1b) Each isometry arises from a unitary matrix of U(dD), with input +and output as shown. +The (blue) ancillary system is initialized at the first step of the sequential generation, transferred +along the process, and accumulated at the final step. (2a) Sequential generation of an isoTNS with physical dimension d and +bond dimension D. In addition to indicating the origin of the process, the diamond also indicates the orthogonality center +of the isoTNS. (2b) Each isometry arises from a unitary matrix of U +� +dD2� +. Ancillary systems are initialized and eventually +accumulated at the boundary of the isoTNS at different steps of the sequential generation. +in random MPS and PEPS from a more operational point +of view. +We introduce families of inhomogeneous ran- +dom tensor network states that arise from a sequential +generation in a quantum computer. Such ensembles are, +by definition, directly connected to the study of quan- +tum circuits with a sequential architecture and random +gates, where each unitary gate is independently cho- +sen randomly according to the uniform (Haar) measure. +In the one-dimensional case, every MPS admits such a +preparation [31], where the bond dimension dictates the +number of overlapping qudits between any two succes- +sive gates. In the two-dimensional setting, our ensemble +can be understood as being uniform over the space of +so-called isometric tensor network states (isoTNS) [32], +which are PEPS with given bond dimension. +In this +case, the resulting family of random circuits is composed +of two-dimensional circuits with local overlapping gates, +each resembling a tile acting on a neighborhood of qu- +dits [33]. +Although isoTNS are a subfamily of PEPS, +they are known to contain a rich variety of strongly cor- +related states, such as topological models [34]. +For the above ensembles, we study the scaling behavior +of the average correlations between two subsystems as a +function of their distance. We then utilize this average +behavior of correlations to deduce the typical case via +concentration inequalities. Instead of using well-known +correlation functions, we perform the analysis using a +measure of correlation arising from the Hilbert–Schmidt +norm. Although in a generic many-body setting such a +measure might have undesirable properties, we show that +it is particularly suited in the context of tensor network +states because it bounds the trace distance as well as all +connected correlation functions. For MPS, we also con- +sider the Rényi-α mutual information. Given a technical +conjecture, we compute the average correlations for all +integer values of α ≥ 1. We then use those results to +retrieve the von Neumann mutual information [35] via +analytic continuation. +First, we confirm analytically the common intuition +that inhomogeneous MPS typically exhibit exponentially +decaying correlations. We show that a single common +correlation length ξ1D persists among different measures +of correlation. We obtain a similar quantitative conclu- +sion for two-dimensional isoTNS, where we observe the +emergence of a different correlation length ξ2D that is also +consistent among different measures of correlation. Both +lengths have a rather weak dependence on the underly- +ing bond dimension D of the tensor network and remain +short-range correlated for all values of D. Surprisingly, +however, ξ1D and ξ2D have exactly opposite behaviors +when D is varied; ξ1D monotonically increases while ξ2D +monotonically decreases. Our findings also establish that +exponentially decaying correlations are typical for the +family of (inhomogeneous) isoTNS, and consequently for +the random states produced by the corresponding quan- +tum circuit architecture. +The paper is structured as follows. In Sec. II, we in- +troduce our families of sequentially generated tensor net- +work states and the main technical tools required to com- +pute their average properties. In Sec. III, we summarize +our results for both MPS and isoTNS. In Secs. IV and V, +we respectively discuss our findings for MPS and isoTNS +in detail. Lastly, we devote Sec. VI to final observations +and potential follow-ups to our work. +II. +PRELIMINARIES +In this section, we introduce the main technical con- +cepts that will be needed throughout the paper. +In +Sec. II A, we review the relevant families of sequentially +generated tensor network states in one and two dimen- +sions. Sec. II B is devoted to the measures of correlation +we are interested to estimate. In Sec. II C, we explain +how to compute averages with respect to the Haar mea- +sure. Lastly, Sec. II D introduces the graphical notation +we will use to present and prove our results. + +3 +A. +Tensor Network States +In one dimension, the preeminent tensor network struc- +ture are matrix product states (MPS) [7]. An n-particle +MPS with open boundary conditions and local (physical) +dimension d is given by +|ψ⟩ = +� +i1,...,in +⟨L|A(1) +i1 · · · A(n) +in |R⟩|i1 · · · in⟩, +(1) +where A(j) ∈ CD×D, |L⟩ ∈ CD is the left boundary con- +dition, and |R⟩ ∈ CD is the right boundary condition. D +is called the bond dimension of the MPS. A commonly +used graphical notation for Eq. (1) is +|ψ⟩ = +, (2) +where vertical (red) legs represent physical space indices +� +Cd� +, and (blue) legs represent bond space indices +� +CD� +. +From its definition, it might not be evident how an +MPS can be prepared because each tensor A does not +necessarily correspond to a physical process. However, +the representation of an MPS in terms of tensors is not +unique. This can be resolved by imposing a convenient +canonical form [36]. Any MPS in such a canonical form +can be seen as a state generated sequentially by applying +unitary matrices U (1), . . . , U (n) ∈ U(dD) to a product +state initialized in |0⟩⊗n for the physical space and |0⟩ +for the bond space [31]. The resulting state is given by +|ψ⟩ = +. +(3) +Note that the final site has dimension dD, while all other +sites have dimension d. As we will see later, the final site +will not play a significant role in our analysis, making its +different dimension not an issue. In Fig. 1 (1a), we sketch +an equivalent representation of sequential generation, in +terms of isometries instead of unitary matrices. +The family of MPS is thus equivalent to states re- +sulting from quantum circuits that have a sequential +architecture and act on input product states. The ar- +chitecture is a consequence of the connectivity of the +MPS network [see Fig. 1 (1a)]. +In this picture, larger +bond dimensions translate to wider gates, each acting on +1+⌈logd(D)⌉qudits. For example, for D = d2 and n = 4, +one has +|ψ⟩ = +. +(4) +Naturally, using this correspondence, all of our results +can be translated to the language of quantum circuits +with the described architecture. +Projected entangled-pair states (PEPS) are the gener- +alization of MPS to two (or more) dimensions [7]. Be- +cause no simple generalization of the sequential gener- +ation of MPS to arbitrary PEPS is known, we restrict +ourselves to the rich family of two-dimensional isometric +tensor network states (isoTNS), which were first defined +in Ref. [32] (see also Ref. [37]). +Much like MPS, isoTNS can be generated sequentially +by applying unitary matrices to a product state initial- +ized in |0⟩⊗mn for the physical space [33], where m de- +notes the number of rows and n the number of columns +of the underlying rectangular lattice. +We will use the +sequential generation sketched in Fig. 1 (2a), which is a +generalization of the one proposed in Ref. [33]. Each box +corresponds to an isometry that arises from a unitary +matrix U (i,j) ∈ U +� +dD2� +. In particular, isometries in the +bulk can be drawn as +, +(5) +as indicated in Fig. 1 (2b). The diamond in Fig. 1 (2a) in- +dicates the so-called orthogonality center of the isoTNS. +Its row and column constitute the orthogonality hyper- +surface, which can be treated like an MPS. That is, if an +operator is supported only on the orthogonality hyper- +surface, its expectation value with respect to the isoTNS +reduces to that of the underlying MPS [32]. Although +isoTNS of a given bond dimension form by definition only +a subclass of PEPS, they are known to contain states with +a rich structure of correlations, such as topological mod- +els [34]. On top, their properties make isoTNS a suitable +candidate for studying correlations analytically, which is +otherwise a generally challenging task in more than one +dimension. +IsoTNS correspond to quantum circuits on a two- +dimensional grid with local overlapping gates, which now +resemble tiles. Increasing bond dimension translates to +larger tile sizes and overlaps, as in the MPS case. The +corresponding architecture is dictated by the connectiv- + +4 +ity of the isoTNS network [see Fig. 1 (2a)], and it is te- +dious (although straightforward), which is why we refer +the reader to Ref. [33] for details. +B. +Quantifying Correlations +Correlations express that knowledge about one sub- +system can convey information about another. +They +are quantified by different measures that frequently arise +from an information-theoretic perspective and are based +on operationally motivated tasks. A prime example is +the von Neumann mutual information [35] +I(A : B) = S(ϱA) + S(ϱB) − S(ϱAB), +(6) +where +S(ϱ) = − tr[ϱ log(ϱ)] +(7) +is the von Neumann entropy. A and B are two disjoint +subsystems of a larger system, and ϱA and ϱB denote the +marginals of ϱAB. The von Neumann mutual information +captures the total (classical and quantum) amount of cor- +relations between A and B, as it is equal to the minimum +rate of randomness required to asymptotically turn ϱAB +into a product state [38]. It is also a non-negative quan- +tity and non-increasing under local operations [35], both +desirable properties for a measure of correlation. +The +latter means that a quantum channel [35] acting on A or +B alone (for example, by discarding part of a subsystem) +cannot increase I(A : B). Unfortunately, the analytical +treatment of the von Neumann mutual information is im- +practical because computing the logarithm of ϱ generally +requires the knowledge of its full spectrum. +To overcome this issue, an alternative is to consider a +particular Rényi-α generalization of the mutual informa- +tion +Iα(A : B) = Sα(ϱA) + Sα(ϱB) − Sα(ϱAB), +(8) +where +Sα(ϱ) = +1 +1 − α log[tr(ϱα)] +(9) +is the Rényi-α entropy. As is apparent from the defini- +tion, for integer values of α, its evaluation is considerably +simpler. The Rényi-α mutual information has been inves- +tigated in the context of conformal field theories [39–41], +free fermions [42], and quantum dynamics [43, 44]. We +will use later that the α → 1 limit of Iα(A : B) recovers +the von Neumann mutual information. The mentioned +positive aspects notwithstanding, unlike the von Neu- +mann mutual information, Eq. (8) does not arise from +a (generalized) divergence [45], and the Rényi-α mutual +information can be negative [46, 47] and increasing under +local operations. It is thus hard to interpret it as a proper +measure of correlation in general. Nevertheless, for cer- +tain families of initial states (see, for example, Ref. [44]) +monotonicity and non-negativity can be restored. Hence- +forth, we will mostly focus on the case of α = 2, but we +will also consider an analytic continuation on positive in- +teger values of α. As we will show, in the present context +of tensor network states, the case of α = 2 appropriately +captures the decay of correlations at large distances be- +tween subsystems A and B with little effort. +In addition to the previous quantities, we would also +like to probe the trace distance +T(A : B) = 1 +2∥ϱAB − ϱA ⊗ ϱB∥1, +(10) +where ∥ · ∥p denotes the Schatten p-norm [48]. For an +operator X, the Schatten p-norm is given by +∥X∥p = tr +�� +X†X +�p/2�1/p +. +(11) +T(A : B) has a well-known operational interpretation, as +it quantifies the optimal distinguishability between ϱAB +and the product of its marginals ϱA⊗ϱB by a two-element +generalized (global) measurement [49]. +Moreover, the +trace distance upper bounds the (properly normalized) +connected correlation function [45]: +T(A : B) ≥ 2|⟨MA ⊗ MB⟩ − ⟨MA⟩⟨MB⟩| +∥MA∥∞∥MB∥∞ +(12) +Although the bound can be tight, the two quantities are +different whenever product measurements are ineffective +in distinguishing ρAB from ρA ⊗ ρB, a fact used in quan- +tum data hiding [50]. +As one expects from its operational interpretation, the +trace distance satisfies the monotonicity property under +local operations [49]. However, T(A : B) is usually hard +to compute exactly. We will now argue that investigating +N(A : B) = ∥ϱAB − ϱA ⊗ ϱB∥2 +2 +(13) +meaningfully probes T(A : B) for tensor network states +with constant bond dimension, all while being much sim- +pler to treat. +In general, for mixed many-body states, the two mea- +sures can have vast disagreement because it holds [48] +that +∥X∥2 ≤ ∥X∥1 ≤ +� +rank(X)∥X∥2. +(14) +Both bounds are tight, and the upper bound is saturated +for X ∝ I. +As such, for arbitrary mixed states of an +exponentially large Hilbert space, the factor rank(X) can +render the upper bound useless. Crucially, in this work +we investigate (random) tensor network states with fixed +bond dimension D. Let ∂R denote the boundary of a +system R and |∂R| its size (number of sites). The ranks +of ϱA and ϱB are respectively upper bounded by D|∂A| +and D|∂B|, and that of ϱAB by D|∂A|+|∂B| [7]. Thus, +rank(ϱAB − ϱA ⊗ ϱB) ≤ 2D|∂A|+|∂B|, +(15) + +5 +yielding the bound +1 +2 +� +N(A : B) ≤ T(A : B) ≤ +� +D|∂A|+|∂B| +2 +� +N(A : B). +(16) +For MPS (one dimension) with connected subsystems +A and B, the bound reads +1 +2 +� +N(A : B) ≤ T(A : B) ≤ D2 +√ +2 +� +N(A : B). +(17) +That is, the bound is independent of the sizes of A and +B, unlike in the generic case of Eq. (14). This suggests +that, for reasonably small bond dimension, N(A : B) is a +reliable probe of correlations [as quantified by T(A : B)] +between subsystems A and B. We will expand on this +point later. +C. +k-Fold Twirl +Let X be an operator acting on (Cq)⊗k. The k-fold +twirl of X with respect to the Haar measure on the uni- +tary group U(q) is defined [51–53] as +T (k) +U +(X) = +� +dU U ⊗kX +� +U †�⊗k. +(18) +One can employ the Schur–Weyl duality for unitary +groups to show [51, 54] that +T (k) +U +(X) = +� +σ,τ∈Sk +Wg +� +στ −1, q +� +P (q) +σ +tr +� +X +� +P (q) +τ +�T � +, +(19) +where +P (q) +π +: v1 ⊗ · · · ⊗ vk �→ vσ−1(1) ⊗ · · · ⊗ vσ−1(k) +(20) +is the representation of π ∈ Sk on (Cq)⊗k, where Sk is +the symmetric group. Wg +� +στ −1, q +� += +� +G−1� +στ [55] is +the Weingarten function, where G ∈ Rk!×k! denotes the +Gram matrix whose entries are given by +Gστ = tr +� +P (q) +σ +� +P (q) +τ +�T � += q#(στ −1). +(21) +Above, #(π) counts the number of cycles in the decom- +position of π ∈ Sk into disjoint cycles. Thus, Wg(π, q) +depends only on the conjugacy class of π [54]. In App. A, +we show how to obtain Eq. (19) from Eq. (18) by using +a result of Ref. [54]. +D. +Graphical Notation +In this section, we introduce the graphical notation +used throughout this paper. To keep the images compact, +we employ the operator-vector correspondence. Let {|i⟩} +denote the standard basis of Cq. +Then, the operator- +vector correspondence [48] is defined by +vec(|i⟩⟨j|) = |i⟩ ⊗ |j⟩ +(22) +and extended linearly to the vector space at large. +Because we consider the standard (product) basis to +be fixed, we do not distinguish between tensors (as mul- +tidimensional arrays) and their basis-independent coun- +terparts (such as vectors and operators). Let X be an +operator acting on (Cq)⊗k. +Using the operator-vector +correspondence, we denote it by += vec(X). +(23) +Note that the orientation of the legs does not have any +meaning in our images. That is, += += += +. +(24) +When we need the transpose of an operator, we will ex- +plicitly use += vec +� +XT � +. +(25) +As such, when we contract two operators X and Y , we +mean the trace of their product: += tr(XY ) +(26) +Let us state the two most prominent operators we will +come across. We will see += vec +� +P (q) +σ +� +, +(27) +where the horizontal (green) leg is permutation valued, +and += vec +� +|0⟩⟨0|⊗k� +. +(28) +Their relevant contractions are += tr +� +|0⟩⟨0|⊗kP (q) +σ +� += 1 +(29) +and += tr +� +P (q) +σ P (q) +τ +� += q#(στ). +(30) +Moving forward, we will not explicitly write the oper- +ator vec as it shall be clear from the context. + +6 +FIG. 2. We investigate average correlations between two sub- +systems A and B as a function of their (horizontal) distance +r. A and B respectively stretch across a and b consecutive +(horizontal) sites. (a) The diamond indicates the origin of the +sequential generation of the MPS. (b) In addition to indicat- +ing the origin of the sequential generation, the diamond also +indicates the orthogonality center of the isoTNS. For now, we +restrict ourselves to A and B that touch the orthogonality +hypersurface and stretch across h consecutive vertical sites. +With the definition of the Weingarten matrix, += Wg +� +στ −1, q +� +, +(31) +we can then write Eq. (19) as +T (k) +U +(X) = +, +(32) +where the contraction of two green legs corresponds to a +summation over the permutations of Sk. +III. +SUMMARY OF RESULTS +In this paper, we analyze the average behavior of cor- +relations in random tensor network states. Through the +average, we obtain conclusions about the generic case. +Our work focuses on the disordered case, that is, the case +where each local tensor is independent. Our setting can +be equivalently understood as an investigation of corre- +lations in states resulting from quantum circuits with a +sequential architecture and random gates. +In one dimension, generic MPS are known to exhibit +exponentially decaying correlations in the translation- +invariant case [56]. This is due to the fact that injectivity +is a generic property [7]. On the other hand, injectivity +alone is not enough to guarantee exponential decay of cor- +relation for an inhomogeneous sequence of tensors. Nev- +ertheless, the exponential decay of correlations is widely +expected to persist without translational invariance but +has never been rigorously studied so far in this setting. +In two (or more) dimensions, the landscape of correla- +tions is much richer. For instance, already in two dimen- +sions, certain PEPS corresponding to thermal states of +classical models are known to exhibit power-law correla- +tions [14]. Moreover, prominent topological states, such +as quantum double models [57] (which include the toric +code) and string-net models [16, 17], admit a description +in terms of PEPS. On the other hand, for translation- +invariant PEPS whose tensors’ entries are drawn from a +Gaussian measure, it is known that correlations typically +decay exponentially [25]. +Computing correlations in higher-dimensional systems +usually poses a significant challenge because they can be +mediated through different paths connecting the two sub- +systems of interest. +Here, we restrict our analysis to +two-dimensional isoTNS. This rich class of tensor net- +work states is relevant in both the analytical and the +numerical context [58–62], all while admitting a simple +physical interpretation through sequential generation [see +Fig. 1 (2a)]. Moreover, its mathematical properties make +the analytical study of correlations in two dimensions +tractable. +For isoTNS, it is expected that correlations between +two subsystems decay exponentially if they are both on +the orthogonality hypersurface because the calculation +reduces to the contraction of an MPS [32]. Nonetheless, +isoTNS can represent a rich variety of topological mod- +els, as all string-net models admit an exact and explicit +description in terms of isoTNS [34] (on the appropriate +underlying lattice). This motivates us to study the typ- +ical behavior of correlations in isoTNS, particularly be- +tween subsystems that extend beyond the orthogonality +hypersurface. +To investigate the decay of correlations in our two fam- +ilies of random tensor network states, one must specify +the ensembles to draw from. +Here we adopt an oper- +ational perspective and relate our measures of random- +ness directly to the sequential generation process. Be- +cause that is defined with respect to isometries, one +can incorporate randomness at the level of the under- +lying unitary matrices. A natural choice is to draw each +unitary from the Haar measure on the appropriate uni- +tary group. This approach was introduced for MPS in +Ref. [18] (see also Ref. [19]), and it can directly be applied +to higher-dimensional tensor network states that admit a +sequential generation, such as isoTNS, yielding normal- +ized states by construction. +Although one can sample +random translation-invariant states with this method, we +investigate the disordered case by drawing each unitary +matrix independently from the Haar measure. +Because we are interested in the decay of correlations, +we focus on computing average correlations between two +subsystems A and B as a function of their distance r. For + +7 +random MPS and isoTNS, we consider subsystems A and +B as sketched in Fig. 2 (a) and Fig. 2 (b), respectively. +In both cases, A and B stretch across a and b consecu- +tive (horizontal) sites. In Fig. 2 (b), A and B touch the +orthogonality hypersurface and stretch across h vertical +sites. We will relax this condition later. +For all of the measures of correlation we study, we find +that the average with respect to the considered ensemble +of states decays exponentially. We formalize this type of +behavior in Definition 1. +Definition 1. Let C(A : B) denote a measure of correla- +tion. We say that the average of C(A : B) with respect to +a given ensemble of random states decays exponentially +if +EC(A : B) = K exp +� +−r +ξ +� ++ O +� +exp +� +− r +χ +�� +, +(33) +where K is constant with respect to r, and ξ > χ is the +average correlation length for C(A : B). +Remarkably, we find that a single average correlation +length persists throughout the different families of mea- +sures of correlation and that it depends only on the un- +derlying spatial dimension. We later pinpoint the origin +of this behavior to the invariance of a spectral gap of the +corresponding family of transfer matrices. For MPS, +ξ = − +� +log +� dD2 − d +d2D2 − 1 +��−1 +≡ ξ1D, +(34) +and for isoTNS, +ξ = − +� +log +�dD3 − dD +d2D4 − 1 +��−1 +≡ ξ2D. +(35) +Note that the average correlation length for MPS co- +incides with that for isoTNS for d → dD. As we will +see later, this seemingly small modification changes the +qualitative behavior substantially. +Before moving to the detailed presentation of our +methods and results, we briefly comment on the consid- +ered measures of correlation and the implications of our +findings, first for MPS and then for isoTNS. +A. +Results in One Dimension (MPS) +In one dimension, we compute the averages of the +Rényi-2 mutual information I2(A : B), the 2-norm ex- +pression N(A : B), and the von Neumann mutual infor- +mation I(A : B) (see Sec. II B for the definitions of the +measures of correlation), with subsystems A and B as +sketched in Fig 2 (a). +We find that the averages decay exponentially as spec- +ified in Definition 1 with the same correlation length ξ1D +(see Results 1, 2, and 3). The derivation for I(A : B) +relies on a technical conjecture (see Conjecture 1), which +we will discuss in detail later. In addition, we show that +the same conjecture is enough to assert that ξ1D is also +the average correlation length for Iα(A : B) for any inte- +ger value of α ≥ 1 (see Corollary 4). +In short, the same average correlation length ξ1D per- +sists across different measures of correlation. +Interest- +ingly, ξ1D depends very weakly on the bond dimension +because, for d, D ≥ 2, +ξ1D = +� +log +� +d +ζ1D(d, D) +��−1 +(36) +with +3 +4 ≤ ζ1D(d, D) < 1 +(37) +is monotonically increasing with D. +In particular, it +holds that limD→∞ ξ1D = 1/ log(d). +Because we are concerned with random tensor network +states, ξ1D is obtained after averaging over realizations. +It is then natural to ask if exponentially decaying corre- +lations are typical and, if so, what is the typical correla- +tion length for an individual realization. This motivates +the investigation of the concentration of the distribution +around its average. To that end, we will show that it is +exponentially unlikely in r that N(A : B) and I(A : B) +decay slower than with ξ1D (see Corollaries 1 and 3). Our +result for N(A : B) allows us to deduce that the average +of the trace distance T(A : B) decays at least exponen- +tially with correlation length ξ ≤ 2ξ1D, and it leads to a +concentration result for T(A : B) (see Corollary 2). +B. +Results in Two Dimensions (isoTNS) +In two dimensions, we compute the averages of the +Rényi-2 mutual information I2(A : B) and the 2-norm +expression N(A : B), where subsystems A and B as +sketched in Fig 2 (b). +As in one dimension, we find that the averages decay +exponentially as specified in Definition 1 with the same +average correlation length ξ2D (see Results 4 and 5). +The correlation length ξ2D displays a qualitatively dif- +ferent dependence on the bond dimension that is albeit +also rather weak. That is, for d, D ≥ 2, +ξ2D = +� +log +� +d +ζ2D(d, D) +��−1 +(38) +with +0 < ζ2D(d, D) ≤ 8 +21. +(39) +In contrast to its one-dimensional counterpart, ξ2D is +a monotonically decreasing function of D. +As such, +perhaps somewhat surprisingly, the largest correlation +length is achieved for D = 2, which is still rapidly de- +caying. +For N(A : B), we can extend the applicability of our +results to any size and shape of subsystems A and B. + +8 +The decay is at least exponential with correlation length +ξ = ξ2D (see Corollary 5). We furthermore prove a con- +centration result for N(A : B) expressing that it is highly +unlikely that N(A : B) decays slower than with ξ2D (see +Corollary 6). This also allows us to draw a similar con- +clusions about the behavior of T(A : B) (see Corollary 7). +IV. +CORRELATIONS IN ONE DIMENSION +In this section, we state and discuss the results for +random MPS summarized in Sec. III A in more detail. +Before doing that, we develop the tools behind our proofs +in Secs. IV A and IV B. In Sec. IV C, we compute the +average of I2(A : B), and in Sec. IV D, we investigate the +decays of N(A : B) and T(A : B). Finally, we discuss +the behavior of I(A : B) in Sec. IV E. +When computing the averages of measures of corre- +lation for random MPS, we will exploit a simplification +with respect to the scenario depicted in Fig. 2 (a). In- +stead of allowing for an arbitrary number of sites be- +fore subsystem A, we prove our statements in the limit +c → ∞. As we show in App. J, this does not constitute +a limitation because the c initial sites do not affect the +decay of correlations and, therefore, neither the average +correlation length ξ1D. Furthermore, we will see that the +f sites after subsystem B do not play a role in the com- +putation of average correlations, as it is expected for any +sequentially generated state. +A. +Transfer Matrices +The key challenge for computing the average of each +measure of correlation will be evaluating multiple expres- +sions of the form +tr +� +PE|ψ⟩⟨ψ|⊗k� +, +(40) +where +P = +� +P (d) +e +�⊗c +⊗ +� +P (d) +α +�⊗a +⊗ +� +P (d) +e +�⊗r +⊗ +� +P (d) +β +�⊗b +⊗ +� +P (d) +e +�⊗(f−1) +⊗ P (dD) +e +. +(41) +The permutation α ∈ Sk acts on the sites comprising +subsystem A, while β ∈ Sk acts on the sites comprising +B. The exact forms of α and β as well as the number of +required replicas k depends on the considered measure of +correlation and will be specified later. It shall also be- +come clear why sites belonging to neither A nor B are +acted upon by the trivial permutation e ∈ Sk. In the fol- +lowing, we show that Eq. (40) for random MPS reduces +to multiplying matrices Tρ ∈ Rk!×k! with ρ ∈ Sk whose +definition will be all but natural. Because their role is +analogous to the known transfer matrices mediating cor- +relations, we will also adopt this term here. +Before introducing the transfer matrices, we must an- +alyze E|ψ⟩⟨ψ|⊗k. +To that end, let us define V (j) = +U (j) ⊗ U (j). Then, by Eq. (3), +|ψ⟩⟨ψ| = +(42) +and +|ψ⟩⟨ψ|⊗k += +. (43) +By computing the k-fold twirl [see Eq. (32)], we obtain +the building block += +� +dU +(44) += +, +(45) +where the (green) dot represents a Kronecker delta on +three permutation indices. Note that we have not drawn +the contraction of a permutation matrix with |0⟩⟨0|⊗k +because it is trivial by Eq. (29). The average of a random +MPS is then given by +E|ψ⟩⟨ψ|⊗k = +. +(46) +We could, in principle, work with the building block +above. However, it is not convenient to have dangling +bond (blue) legs whose dimension grows with D. +By +cutting permutation-valued (green) legs instead, we ob- +tain a building block with fixed dimension for fixed k. +With that building block, the average of a random MPS +is given by +E|ψ⟩⟨ψ|⊗k = +. +(47) +The entries of the initial vector ⟨Ik| ∈ Rk! are given by += += 1, +(48) + +9 +where we have used Eq. (29). The tensors in the bulk are +given via += +, +(49) +and the final tensor is given via += +. +(50) +Computing an expression of the form of Eq. (40) cor- +responds to contracting each S with some P (d) +ρ +, which +leads us to the promised definition of a transfer matrix +Tρ ∈ Rk!×k!. Using Eq. (30), its entries are given by += += +(51) += +� +σ∈Sk +Wg +� +στ −1, dD +� +d#(σρ)D#(σθ−1). +(52) +We define Tρ with respect to the basis defined by the map +si �→ ei, where si is the ith element of Sk = {s1, . . . , sk!}, +and {ei} is the standard basis of Rk!. +In App. D, we +find that Tρ is block triangular if the elements of Sk are +ordered in a certain way. +As alluded to in Eq. (41), the final tensor S′ will be +contracted with the trivial permutation e ∈ Sk in our +computations. The final vector |Fk⟩ ∈ Rk! is thus defined +via += += +(53) += +� +σ∈Sk +Wg +� +στ −1, dD +� +(dD)#(σe) = δeτ , +(54) +where we have used the definition of the Weingarten func- +tion. +Using the definitions of Te and |Fk⟩, it is easy to con- +firm that Te|Fk⟩ = |Fk⟩. Graphically, this implies the +simplification += +. +(55) +From this, it also follows that E|ψ⟩⟨ψ|⊗k is properly nor- +malized: +tr +� +E|ψ⟩⟨ψ|⊗k� += ⟨Ik|T n−1 +e +|Fk⟩ = ⟨Ik|Fk⟩ = 1 +(56) +With what we have laid out above, we can write +Eq. (40) in terms of transfer matrices: +tr +� +PE|ψ⟩⟨ψ|⊗k� += ⟨Ik|T c +e T a +αT r +e T b +β|Fk⟩ +(57) +We provide a simple Mathematica package [63] that +defines Tρ with ρ ∈ Sk for k ∈ {1, . . . , 20} according to +Eq. (52). +The package relies on the package provided +by the authors of Ref. [64] for evaluating the Weingarten +function. +B. +Estimating the Decay of Correlations +The decay of the average of each measure of correla- +tion is necessarily determined by the r sites separating +subsystems A and B. As we will see in the following sec- +tions, this will, for each measure, translate to a simple +statement in terms of the just-defined transfer matrices. +In particular, we will find that the decay of correlations +is determined by T r +e with e ∈ Sk. Taking this as a fact +for now, we connect the decay of correlations with the +spectrum of Te. +The spectrum of Te depends on k because k determines +its dimension and entries. +Still, we can make general +statements about Te for any k ≥ 2. In particular, we will +prove the following statements in Apps. B and C. +Proposition 1. The eigenvalues of Te with e ∈ Sk are +non-negative for any k ≥ 2. +Proposition 2. Te with e ∈ Sk is diagonalizable for any +k ≥ 2. +Proposition 3. Let λ1 > λ2 > · · · ≥ 0 denote the dis- +tinct eigenvalues of Te with e ∈ Sk. Then, λ1 = 1 and it +is non-degenerate for any k ≥ 2 if d ≥ 2. +Given the statements above, we can expand T r +e as +T r +e = |R1⟩⟨L1| + λr +2 +w2 +� +µ=1 +|R(µ) +2 ⟩⟨L(µ) +2 | + O(λr +3), +(58) +where |R(µ) +i +⟩ denotes the µth right eigenvector corre- +sponding to λi, ⟨L(µ) +i +| denotes the µth left eigenvector +corresponding to λi, and wi denotes the degeneracy of +λi. +The asymptotic decay of correlations is thus deter- +mined by the subleading eigenvalue λ2 of Te, and the +average correlation length is given by +ξ = − +1 +log(|λ2|). +(59) +The argument behind this is similar to one known +from the analysis of correlations in translation-invariant +MPS [56, 65], where the decay is determined by the sub- +leading eigenvalue of the relevant transfer matrix. + +10 +C. +Rényi-2 Mutual Information +We start our analysis of correlations in random MPS +with the simplest case, namely the computation of the +average of the Rényi-2 mutual information +I2(A : B) = log +� +tr +� +ϱ2 +AB +�� +− log +� +tr +� +ϱ2 +A +�� +− log +� +tr +� +ϱ2 +B +�� +. +(60) +The analytical treatment turns out to be comparatively +simple if one assumes that E log(X) = log(EX), as is +frequently done in this context [66–68]. We will not make +this assumption further below when we study the von +Neumann mutual information. The analysis there will +require transfer matrices Tρ with ρ ∈ Sk for all k ≥ 2, +while ρ ∈ S2 will suffice here because only the averages +of purities are needed. +Our first result is summarized +below. +Result 1. The average of I2(A : B) with respect to +the random MPS ensemble and subsystems A and B as +sketched in Fig. 2 (a) decays exponentially as specified +in Definition 1 with the average correlation length ξ1D +defined in Eq. (34). +Proof. We split the proof into four steps. The exact same +structure will also appear in the proofs for the other mea- +sures of correlation. Thus, this proof serves as the sim- +plest example and a point of reference for later proofs. +Step 1. We rewrite EI2(A : B) in terms of expressions +of the form of Eq. (40). To that end, we make the as- +sumption that E log(X) = log(EX). Then, +EI2(A : B) = log +� +E tr +� +ϱ2 +AB +�� +− log +� +E tr +� +ϱ2 +A +�� +− log +� +E tr +� +ϱ2 +B +�� +. +(61) +E tr +� +ϱ2 +A +� +, E tr +� +ϱ2 +B +� +, and E tr +� +ϱ2 +AB +� +can already be written +in the desired form. For example, +E tr +� +ϱ2 +AB +� += tr +� +PE|ψ⟩⟨ψ|⊗2� +(62) +with +P = +� +P (d) +e +�⊗c +⊗ +� +P (d) +(12) +�⊗a +⊗ +� +P (d) +e +�⊗r +⊗ +� +P (d) +(12) +�⊗b +⊗ +� +P (d) +e +�⊗(f−1) +⊗ P (dD) +e +. +(63) +Step 2. We express EI2(A : B) in terms of the transfer +matrices defined in Sec. IV A. Given the previous step, it +is easy to confirm that +EI2(A : B) = log +� +⟨I2|T c +e T a +(12)T r +e T b +(12)|F2⟩ +� +− log +� +⟨I2|T c +e T a +(12)|F2⟩ +� +− log +� +⟨I2|T c+a+r +e +T b +(12)|F2⟩ +� +. +(64) +Step 3. We expand EI2(A : B) in terms of the spectrum +of Te with e ∈ S2 [see Eq. (58)]. +Using the relevant +expressions for the Weingarten function, it is evident [69] +that +Te = +� +� +� +� +1 d2D − D +d2D2 − 1 +0 +dD2 − d +d2D2 − 1 +� +� +� +� +(65) +is diagonalizable with +λ1 = 1 +and +λ2 = dD2 − d +d2D2 − 1. +(66) +Expanding T c +e and taking the limit c → ∞ yields +EI2(A : B) = log +� +⟨L1|T a +(12)T r +e T b +(12)|F2⟩ +� +− log +� +⟨L1|T a +(12)|F2⟩ +� +− log +� +⟨L1|T b +(12)|F2⟩ +� +, +(67) +where we have used that ⟨I2|R1⟩ = 1. After expanding +also T r +e and using that |F2⟩ = |R1⟩, we have +EI2(A : B) = log +� +⟨L1|T a +(12)|R1⟩⟨L1|T b +(12)|R1⟩ ++ λr +2⟨L1|T a +(12)|R2⟩⟨L2|T b +(12)|R1⟩ +� +− log +� +⟨L1|T a +(12)|R1⟩ +� +− log +� +⟨L1|T b +(12)|R1⟩ +� +. +(68) +Step 4. Finally, we can write EI2(A : B) in the form of +Definition 1. That is, +EI2(A : B) += log +� +1 + λr +2 +⟨L1|T a +(12)|R2⟩⟨L2|T b +(12)|R1⟩ +⟨L1|T a +(12)|R1⟩⟨L1|T b +(12)|R1⟩ +� +(69) += λr +2 +⟨L1|T a +(12)|R2⟩⟨L2|T b +(12)|R1⟩ +⟨L1|T a +(12)|R1⟩⟨L1|T b +(12)|R1⟩ + O +� +λ2r +2 +� +(70) +≡ K exp +� +−r +ξ +� ++ O +� +exp +� +−2r +ξ +�� +, +(71) +where +K = +⟨L1|T a +(12)|R2⟩⟨L2|T b +(12)|R1⟩ +⟨L1|T a +(12)|R1⟩⟨L1|T b +(12)|R1⟩ +(72) +and +ξ = − +1 +log(λ2) = − +� +log +� dD2 − d +d2D2 − 1 +��−1 += ξ1D. +(73) +This concludes the proof. + +11 +As we discussed earlier, the Rényi-2 mutual informa- +tion is lacking many of the desirable properties that a +sound measure of correlation ought to fulfill. +On top, +our computation simplifies considerably because we are +using the assumption that E log(X) = log(EX), which +amounts to ignoring statistical fluctuations in the dif- +ferent realizations. In the following section, we will see +that N(A : B) decays exponentially with the same av- +erage correlation length ξ1D. We will furthermore show +that N(A : B) concentrates around its average, provid- +ing evidence that fluctuations can be safely ignored in +our context. +D. +Trace Distance and 2-Norm +In this section, we investigate average correlations as +quantified by the trace distance T(A : B). As anticipated +in Sec. II B, this is a challenging task. However, as laid +out there, the 2-norm expression N(A : B) reliably esti- +mates T(A : B) for the case of random MPS. Hence, we +compute the average of +N(A : B) = ∥ϱAB − ϱA ⊗ ϱB∥2 +2 +(74) += tr +� +ϱ2 +AB +� ++ tr +� +ϱ2 +A +� +tr +� +ϱ2 +B +� +− 2 tr[ϱAB(ϱA ⊗ ϱB)]. +(75) +Because of its connection to the Hilbert–Schmidt in- +ner product, the average of N(A : B) can be computed +without any simplifying assumptions. Making use of the +transfer-matrix techniques introduced above, we prove +the following result. +Result 2. The average of N(A : B) with respect to +the random MPS ensemble and subsystems A and B as +sketched in Fig. 2 (a) decays exponentially as specified +in Definition 1 with the average correlation length ξ1D +defined in Eq. (34). +Sketch of proof. The proof follows the same procedure as +that of Result 1. +Here, we sketch the main steps and +refer to App. H for more details. +In Step 1, we write EN(A : B) in terms of expressions +of the form of Eq. (40). The second summand in Eq. (75) +requires permutations of S4 because +E tr +� +ϱ2 +A +� +tr +� +ϱ2 +B +� += tr +� +PE|ψ⟩⟨ψ|⊗4� +(76) +with +P = +� +P (d) +e +�⊗c +⊗ +� +P (d) +(12) +�⊗a +⊗ +� +P (d) +e +�⊗r +⊗ +� +P (d) +(34) +�⊗b +⊗ +� +P (d) +e +�⊗(f−1) +⊗ P (dD) +e +. +(77) +The first summand and the third summand respectively +require only permutations of S2 and S3. In Step 2, we +thus write EN(A : B) in terms of transfer matrices Tρ +with ρ ∈ S4. +This means that the average correlation length is de- +termined by the subleading eigenvalue of Te with e ∈ S4. +Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of +Te. In Step 3, we find that +λ1 = 1 +and +λ2 = dD2 − d +d2D2 − 1, +(78) +just like for Te with e ∈ S2. +The former is non- +degenerate, while the degeneracy of the latter is given +by the number of transposition in S4, +w2 = +�4 +2 +� += 6. +(79) +Thus, the average correlation length for N(A : B) co- +incides with that for I2(A : B), as we conclude in +Step 4. +The above result establishes the exponential decay of +the average. However, one is usually interested in know- +ing if typical instances are expected to have the same +exponential decay. +This can be easily established by +Markov’s inequality because N(A : B) is non-negative +and its average decays to zero as a function of the dis- +tance r. +Corollary 1. For subsystems A and B as sketched in +Fig. 2 (a), sufficiently large r, and all 0 < ε < 1, the +random MPS ensemble satisfies +Pr +� +N(A : B) ≥ K exp +� +−(1 − ε)r +ξ1D +�� +≤ exp +� +− εr +ξ1D +� +, +(80) +where K is constant with respect to r. +Proof. By Result 2, for sufficiently large r, we can bound +EN(A : B) ≤ K exp(−r/ξ1D). +Because N(A : B) is +non-negative, by Markov’s inequality, we have, for η > 0, +Pr +� +N(A : B) ≥ ηK exp +� +− r +ξ1D +�� +≤ Pr[N(A : B) ≥ ηEN(A : B)] +(81) +≤ 1 +η . +(82) +The result follows with η = exp(εr/ξ1D). +The above corollary reflects that it is exponentially +unlikely in r that N(A : B) decays slower than with the +average correlation length ξ1D. Because we have already +established that the average case exhibits an exponential +decay with correlation length ξ1D, the average case is also +typical. +By combining the above result with Eq. (17), we can +now also bound the correlation length for T(A : B). + +12 +1.4 +0.9 +ξ +6 +8 +10 +12 +14 +D +0.6 +0.7 +d = 2 +d = 3 +d = 4 +d = 5 +FIG. 3. Numerically obtained average correlation length ξ for +different d and D. The data points are obtained by fitting the +average value of I(A : B) against r ∈ {5, 7, 9, 11, 13, 15} for +a = b = 1. The sample size of 10 000 suffices for the error bars +to lie within the plot points. The opaque curves correspond +to ξ1D [see Eq. (34)] +Corollary 2. For subsystems A and B as sketched in +Fig. 2 (a), sufficiently large r, and all 0 < ε < 1, the +random MPS ensemble satisfies +Pr +� +T(A : B) ≥ K exp +� +−(1 − ε)r +2ξ1D +�� +≤ exp +� +− εr +ξ1D +� +, +(83) +where K is constant with respect to r. +Proof. It holds that +ET(A : B) ≤ +� +E[T(A : B)]2 +(84) +≤ D2 +2 +� +EN(A : B) +(85) +≤ K exp +� +− +r +2ξ1D +� +, +(86) +where, in the last line, we have assumed r to be suffi- +ciently large. The result follows as in the proof of Corol- +lary 1. +Thus, with overwhelming probability, correlations as +quantified by T(A : B) decay exponentially with ξ ≤ +2ξ1D. +E. +Von Neumann Mutual Information +The fact that I2(A : B) and N(A : B) have the same +average correlation length ξ1D motivates the question +whether other measures of correlation behave similarly. +In this section, we will provide compelling evidence that +ξ1D is indeed the average correlation length also for the +von Neumann mutual information I(A : B). +We start by numerically investigating the behavior of +I(A : B) for random MPS. We have generated MPS ac- +cording to our measure, computed the average of I(A : +B), and extracted the average correlation length from +fits. As Fig. 3 shows, the numerically obtained average +correlation length coincides well with ξ1D. It should be +noted that we have set c = 0 for our numerical analysis. +We discuss in App. J why this does not affect the aver- +age correlation length. For more details on our numerical +analysis, see App. K. +We now turn to the analytical computation of the av- +erage of +I(A : B) = tr[ρAB log(ρAB)] +− tr[ρA log(ρA)] +− tr[ρB log(ρB)]. +(87) +To be able to make use of the transfer-matrix tech- +niques introduced above, we employ two replica tricks to +write EI(A : B) in terms of expressions of the form of +Eq. (40). +First, we write S(ρ) as the α → 1 limit of +Sα(ρ): +S(ρ) = lim +α→1 +1 +1 − α log[tr(ϱα)] +(88) +Second, instead of assuming again that E log(X) = +log(EX), we use +E log(X) = lim +v→0 +1 +v log(EXv). +(89) +We are thus dealing with expressions of the form +tr(PE|ψ⟩⟨ψ|⊗vα), which require transfer matrices Tρ with +ρ ∈ Svα (see App. I). This means that knowing the spec- +trum of Te with e ∈ Svα for all vα ≥ 2 allows us to draw +conclusions about the decay of the average of I(A : B). +While this is in principle a daunting task, we are able +to prove several properties of the transfer matrix Te with +e ∈ Sk for any k ≥ 2 (see Propositions 1, 2, and 3). +In App. D, we furthermore show that Te with e ∈ Sk +has eigenvalue +µ2 = dD2 − d +d2D2 − 1. +(90) +for any k ≥ 2. Its degeneracy is at least +v2 = +�k +2 +� +, +(91) +the number of transpositions in Sk. We conjecture that +µ2 is the subleading eigenvalue of Te with e ∈ Sk for +any k ≥ 2 and that it has degeneracy v2. +We know +this conjecture to hold for k ∈ {2, 3, 4}, and we have +numerical evidence suggesting so for k ∈ {5, 6, 7} [63]. +We were not able to prove the statement outright, but +in the following we argue that it is the only missing step +to show that ξ1D is the average correlation length for +I(A : B). Let us state the conjecture formally below. + +13 +Conjecture 1. Let λ1 > λ2 > · · · ≥ 0 denote the dis- +tinct eigenvalues of Te with e ∈ Sk. Then, λ2 = µ2 with +degeneracy w2 = v2 for any k ≥ 2. +If Conjecture 1 holds, the properties of Te with e ∈ Sva +that are relevant for determining the decay of correlations +are independent of vα. +In App. I, we argue that the +replica limit does not affect this and prove the following +result. +Result 3. If Conjecture 1 holds, the average of I(A : B) +with respect to the random MPS ensemble and subsystems +A and B as sketched in Fig. 2 (a) decays exponentially +as specified in Definition 1 with the average correlation +length ξ1D defined in Eq. (34). +We provide a concentration result also for I(A : B). +The statement and its proof are identical to that for +N(A : B). It is exponentially unlikely in r that I(A : B) +decays slower than with the average correlation length +ξ1D, and the average case is also typical. +Corollary 3. If Conjecture 1 holds, for subsystems A +and B as sketched in Fig. 2 (a), sufficiently large r, and +all 0 < ε < 1, the random MPS ensemble satisfies +Pr +� +I(A : B) ≥ K exp +� +−(1 − ε)r +ξ +�� +≤ exp +� +−εr +ξ +� +, +(92) +where K is constant with respect to r. +Another corollary of Result 3 is that ξ1D is also the +average correlation length for Iα(A : B) for any integer +value of α ≥ 1. We prove also this statement in App. I. +Corollary 4. If Conjecture 1 holds, for any integer value +of α ≥ 1, the average of Iα(A : B) with respect to the ran- +dom MPS ensemble and subsystems A and B as sketched +in Fig. 2 (a) decays exponentially as specified in Defini- +tion 1 with the average correlation length ξ1D defined in +Eq. (34). +Finally, let us summarize the reason behind the per- +sistent appearance of the average correlation length ξ1D. +In all of the examined cases, the asymptotic behavior of +the correlations was determined by the asymptotic de- +cay of T r +e . Although the transfer matrix Te with e ∈ Sk +does depend on the number of replicas k, its asymptotic +decay does not (given Conjecture 1), resulting in a com- +mon average correlation length ξ1D across measures of +correlation with different complexity. +V. +CORRELATIONS IN TWO DIMENSIONS +In this section, we state and discuss in more detail +the results for random isoTNS summarized in Sec. III B. +In Sec. V A, we develop the two-dimensional analog to +the transfer matrices introduced in Sec. IV A, the tool +behind our proofs. In Sec. V C, we compute the average +FIG. 4. We investigate average correlations in random isoTNS +between two subsystems A and B as a function of their hori- +zontal distance r. A and B respectively stretch across a and +b consecutive horizontal sites. In addition to indicating the +origin of the sequential generation, the diamond also indicates +the orthogonality center of the isoTNS. (a) For Results 4 and +5, we consider A and B that touch the orthogonality hyper- +surface and stretch across h consecutive vertical sites. (b) We +will provide an additional result for the 2-norm expression +N(A : B) for arbitrary (but fixed) A and B that do not need +to touch the orthogonality hypersurface. +of I2(A : B), and in Sec. V D, we investigate the decays +of N(A : B) and T(A : B). +As in one dimension, we prove our statements in the +limit c → ∞. +We show the exponential decay of the average for each +considered measure of correlation and subsystems A and +B as sketched in Fig. 4 (a) as a function of the distance r +between A and B. In particular, we prove that I2(A : B) +and N(A : B) have a common average correlation length +that is independent of the sizes of A and B. Moreover, +thanks to the more amenable properties of N(A : B), we +are able to show that average correlations decay expo- +nentially also for subsystems A and B that do not have +to touch the orthogonality hypersurface [see Fig. 4 (b)]. +A. +Transfer Matrices +As in one dimension, computing the average of each +measure of correlation will involve computing multiple + +14 +terms of the form +tr +� +PE|ψ⟩⟨ψ|⊗k� +, +(93) +where P +has a similar tensor product structure as +Eq. (41), adapted to the two-dimensional setting con- +sidered here. The number of required replicas k again +depends on the considered measure of correlation. +We will thus need a two-dimensional analog to the +transfer matrices introduced in Sec. IV A. In contrast to +the one-dimensional case, the size of the resulting transfer +matrices will also depend on the geometry of the consid- +ered subsystems, making their analysis much more chal- +lenging. However, the procedure of defining the tensors is +similar to that in one dimension. We provide an overview +here and refer to App. L for more details. +We define V (i,j) = U (i,j) ⊗ U (i,j), where U (i,j) ∈ +U +� +dD2� +is the unitary matrix depicted in Fig. 1 (2b). +By computing the k-fold twirl [see Eq. (32)], we obtain +the building block += +� +dU +(94) += +. +(95) +As in one dimension, it is convenient to define a build- +ing block for which the contraction of bond (blue) legs is +implicit. Analogously to the one-dimensional case, this +results in a tensor with only permutation-valued (green) +legs. As we show in App. L, the resulting bulk tensor is +given via += +. +(96) +For the sake of brevity, the expressions for the boundary +tensors are stated in the appendix. +Computing expressions of the form of Eq. (93) corre- +sponds to contracting each S with some P (d) +ρ +. The entries +of the resulting bulk tensor Tρ ∈ Rk!×k!×k!×k! are given +by += += +(97) += +� +σ∈Sk +Wg +� +στ −1, dD2� +d#(σρ)D#(σθ−1)D#(σν−1). +(98) +In our computations, the site in the top right corner +belongs to neither A nor B. It is thus acted upon by the +trivial permutation e ∈ Sk. As we show in App. L, the +corresponding tensor is given by Te = |Fk⟩⟨Fk|. +In App. L, we furthermore show that a property similar +to Eq. (55) also holds for random isoTNS. That is, tensors +corresponding to the trivial permutation e ∈ Sk on the +boundary simplify. For example, += +. +(99) +As for the one-dimensional case, the proper normaliza- +tion of E|ψ⟩⟨ψ|⊗k follows directly from this property. +Instead of thinking in terms of contractions of two- +dimensional tensor networks, it will later prove beneficial +to think again in terms of multiplications of matrices. To +that end, for any height h of the subsystems A and B [see. +Fig. 4 (a)], we define += +(100) + +15 +as well as += +and += +. +(101) +For subsystems A and B as sketched in Fig. 4 (a), we +can now write Eq. (93) in terms of transfer matrices: +tr +� +PE|ψ⟩⟨ψ|⊗k� += ⟨Ik|T c +e T a +α T r +e T b +β |Fk⟩ +(102) +We provide an additional Mathematica package [63] +that defines Tρ with ρ ∈ Sk for k ∈ {1, . . . , 20} according +to Eq. (98). Once again, the package relies on the one +provided by the authors of Ref. [64] for evaluating the +Weingarten function. +B. +Estimating the Decay of Correlations +Eq. (102) implies that, for subsystems A and B as +sketched in Fig. 4 (a), the decay of the average of each +measure of correlation will again reduce to a statement +in terms of transfer matrices. In particular, the decay +will be determined by the spectrum of Te with e ∈ Sk. +Notice that, in addition to a dependence on k, the form +and properties of Te now depend also on the height h of +the subsystems A and B [see. Fig. 4 (a)]. However, as +we prove in the following sections, its two leading eigen- +values are independent of h for at least k = 2 and k = 4. +Crucially, this will allow us to make statements about +the decay of the averages of I2(A : B) and N(A : B) for +arbitrary h. +Note that we could, in principle, investigate vertical +separation instead of horizontal separation because += +. +(103) +The underlying exchange of indices does not affect the +spectrum of the relevant identity transfer matrix and thus +neither the average correlation length. This reflects the +fact that the sequential generation procedure is symmet- +ric in the horizontal and vertical spatial directions. +C. +Rényi-2 Mutual Information +In this section, we compute the average of the Rényi-2 +mutual information I2(A : B) for random isoTNS that +are generated as sketched in Fig. 1 (2a). Subsystems A +and B are defined in Fig. 4 (a). +As in one dimension, we will make the assumption that +E log(X) = log(EX) (see Sec. IV C). Step 1 of the proof +of Result 4 (see App. O) is thus all but identical to Step 1 +of the proof of Result 1. +Also Step 2 is largely analogous. To see this, let us +take E tr +� +ϱ2 +A +� +as an example. With A and B as defined +in Fig. 4 (a) and x = (12), +E tr +� +ϱ2 +A +� += +(104) += +(105) += ⟨I2|T c +e T a +x |F2⟩. +(106) +The resulting expression for EI2(A : B), +EI2(A : B) = log +� +⟨I2|T c +e T a +x T r +e T b +x |F2⟩ +� +− log +� +⟨I2|T c +e T a +x |F2⟩ +� +− log +� +⟨I2|T c+a+r +e +T b +x |F2⟩ +� +, +(107) +resembles Eq. (64) closely. +The remaining technical challenge is the analysis of +the spectrum of Te with e ∈ S2. Let λ1 > λ2 > · · · ≥ 0 +denote its distinct eigenvalues. We show in App. M that +λ1 = 1 +and +λ2 = dD3 − dD +d2D4 − 1 +(108) +and that both eigenvalues are non-degenerate. +In the +proof, that holds for any h, we map the contraction of +tensors defining Te with e ∈ S2 to a multiplication of ma- +trices. Using the Weingarten calculus, we show that Te +is upper block triangular, a property that simplifies the +analysis of its spectrum. The main difficulty is then to +prove that the specified λ2 is indeed the subleading eigen- +value for all h. We do this by exploiting substochasticity. +Following the same reasoning as before, we find that +the average of I2(A : B) decays exponentially as specified +in Definition 1. We prove this result in App. O. +Result 4. The average of I2(A : B) with respect to the +random isoTNS ensemble and subsystems A and B as +sketched in Fig. 4 (a) decays exponentially as specified +in Definition 1 with the average correlation length ξ2D +defined in Eq. (35). + +16 +D. +Trace Distance and 2-Norm +In this section, we show the exponential decay of the +average of N(A : B) for random isoTNS. As for the one- +dimensional case, we do that to eventually make conclu- +sions about the behavior of the trace distance T(A : B). +While it is not trivial to compute the average of +N(A : B) with respect to the random isoTNS ensemble +and subsystems A and B as sketched in Fig. 4 (a), the +computation follows along the lines of what we have laid +out in Sec. IV D. In particular, we find that the decay of +the average of N(A : B) is determined by the spectrum +of the transfer matrix Te with e ∈ S4. +Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues +of Te with e ∈ S4. As we show in App. N, for any h, +λ1 = 1 +and +λ2 = dD3 − dD +d2D4 − 1 . +(109) +As in one dimension, the former is non-degenerate, while +the degeneracy of the latter is given by the number of +transpositions in S4, +w2 = +�4 +2 +� += 6. +(110) +While the analysis of the spectrum is, in principle, similar +to that of the spectrum of Te with e ∈ S2, it is consider- +ably more technical. This is largely due to the fact that +the matrices whose multiplication defines Te with e ∈ S4 +are significantly more complex. Still, we find also Te with +e ∈ S4 to be upper block triangular, allowing us to show +that the specified λ2 is indeed the subleading eigenvalue. +Given λ2 of Te with e ∈ S4 and the arguments devel- +oped in Sec. IV D, we can state the first result of this +section, which we prove in App. P. +Result 5. The average of N(A : B) with respect to the +random isoTNS ensemble and subsystems A and B as +sketched in Fig. 4 (a) decays exponentially as specified +in Definition 1 with the average correlation length ξ2D +defined in Eq. (35). +We now turn to the case of correlations in isoTNS with +arbitrary (but fixed) subsystems A and B [see Fig. 4 (b)]. +The decay of the average of N(A : B) for arbitrary A +and B can be bounded by employing the fact that the +Schatten 2-norm satisfies [70] +∥trB(XAB)∥2 ≤ +� +dim(B)∥XAB∥2, +(111) +where XAB is any bipartite operator and dim(B) is the +dimension of the Hilbert space that is traced out. This +means that +N(A : B) ≤ d|AC|+|BC|N(A′ : B′), +(112) +where A is now an arbitrary subsystem, A′ is its (min- +imal) enclosing rectangle that touches the hypersurface, +and AC = A′ − A. B′ and BC are defined analogously +for B. +Using the statement above, we can bound the decay +of the average of N(A : B) for arbitrary A and B as a +straightforward corollary of Result 5. We stress that we +consider regime in which the distance r between A and +B grows. +Corollary 5. For arbitrary subsystems A and B, the +average of N(A : B) with respect to the random isoTNS +ensemble decays as +N(A : B) = O +� +exp +� +− r +ξ2D +�� +, +(113) +where the average correlation length ξ2D is defined in +Eq. (35). +Finally, we state a concentration result for N(A : B), +which, in combination with Eq. (16), also allows us to +draw conclusions about the typical behavior of T(A : B). +As before, we consider arbitrary (but fixed) subsystems +A and B [see Fig. 4 (b)]. +Corollary 6. For arbitrary subsystems A and B, suffi- +ciently large r, and all 0 < ε < 1, the random isoTNS +ensemble satisfies +Pr +� +N(A : B) ≥ K exp +� +−(1 − ε)r +ξ2D +�� +≤ exp +� +− εr +ξ2D +� +, +(114) +where K is constant with respect to r and the average +correlation length ξ2D is defined in Eq. (35). +Corollary 7. For arbitrary subsystems A and B, suffi- +ciently large r, and all 0 < ε < 1, the random isoTNS +ensemble satisfies +Pr +� +T(A : B) ≥ K exp +� +−(1 − ε)r +2ξ2D +�� +≤ exp +� +− εr +ξ2D +� +(115) +where K is constant with respect to r and the average +correlation length ξ2D is defined in Eq. (35). +The proofs and discussions of these results are identical +to their one-dimensional counterparts. +The tools we have developed in this and the previous +two sections should, in principle, allow us to make state- +ments also about the decay of the average of the von +Neumann mutual information I(A : B) with respect to +the random isoTNS ensemble. As in one dimension, we +would need to investigate the spectrum of Te with e ∈ Sk +for all k ≥ 2. However, as the analysis of the spectrum of +Te is already quite technical for e ∈ S4 with our methods, +we refrain from tackling the spectrum for k ≥ 5 here. +VI. +CONCLUSION +We have investigated the average behavior of correla- +tions between two distant subsystems A and B for en- +sembles of random MPS and isoTNS. As measures of + +17 +correlation, we have considered the Rényi-α mutual in- +formation, a measure arising from the Hilbert–Schmidt +norm, the trace distance, and the von Neumann mutual +information. +We have shown that the average of each +considered measure exhibits an exponential decay. Our +results can equivalently be seen as describing states re- +sulting from quantum circuits with a sequential architec- +ture and Haar random gates. +By leveraging the Weingarten calculus, we have devel- +oped a mathematical framework that allows to infer the +average correlation length from the subleading eigenvalue +of an appropriately defined transfer matrix. +We have +computed the averages of the Rényi-α mutual informa- +tion and the measure arising from the Hilbert–Schmidt +norm to show the emergence of an average correlation +length that only depends on the underlying spatial di- +mension but not the considered measure. In particular, +the average correlation length for random MPS increases +weakly with the bond dimension D and converges rapidly +(as D grows) to the value 1/ log(d), where d is the phys- +ical dimension. On the contrary, the average correlation +length for random isoTNS, while still depending on the +bond dimension only weakly, decreases with D. Surpris- +ingly, the highest average correlation length for random +isoTNS is achieved with the lowest non-trivial bond di- +mension (D = 2). +Using elementary concentration results, we have fur- +thermore deduced the typical behavior of the measure +arising from the Hilbert–Schmidt, which has in turn al- +lowed us to make similar statements about the trace dis- +tance. +For MPS, we have been able to give strong indications +that the universal correlation length applies also to the +von Neumann mutual information, and also any Rényi- +α mutual information for integer values of α ≥ 1. +It +would be interesting to prove this behavior rigorously, +and also investigate its validity for the isoTNS case. An- +other possible future direction would be to study average +correlations in more general random PEPS, beyond the +class of isoTNS, as well as other types of quantum circuit +architectures. +ACKNOWLEDGMENTS +We thank Ignacio Cirac and Philippe Faist for fruitful +discussions. 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Stat. 14, 631 (2017), arXiv:1701.04493 +[math.CO]. +[77] C. Akers, T. Faulkner, S. Lin, and P. Rath, Reflected +entropy in random tensor networks, J. High Energy Phys. +2022, 162, arXiv:2112.09122 [hep-th]. +[78] P. Biane, Some properties of crossings and partitions, +Discrete Mathematics 175, 41 (1997). +[79] T. Castellani and A. Cavagna, Spin-glass theory for +pedestrians, +J. +Stat. +Mech. +2005, +P05012 +(2005), +arXiv:cond-mat/0505032. +Appendix A: k-Fold Twirl +In this appendix, we present some additional details on the k-fold twirl. In particular, we go from Eq. (18) to +Eq. (19). To do this, we will need a result of Ref. [54], which appears as Corollary 2.4 in Ref. [51]. We state it without +proof as Lemma 1. +Lemma 1. Let k be a positive integer, and (i1, . . . , ik), (j1, . . . , jk), (ℓ1, . . . , ℓk), and (m1, . . . , mk) be k-tuples of +positive integers. Then, +� +dU Ui1ji · · · UikjkUm1ℓ1 · · · Umkℓk = +� +σ,τ∈Sk +Wg +� +σ−1τ, q +� +δi1mσ(1) · · · δikmσ(k)δj1ℓτ(1) · · · δjklτ(k), +(A1) +where the integration is with respect to the Haar measure on the unitary group U(q). + +20 +By Lemma 1, +� +T (k) +U +(X) +� +i1···ikm1···mk += +� +j1,...,jk +ℓ1,...,ℓk +� +dU Ui1ji · · · UikjkXj1···jkℓ1···ℓkUm1ℓ1 · · · Umkℓk +(A2) += +� +j1,...,jk +ℓ1,...,ℓk +� +σ,τ∈Sk +Wg +� +σ−1τ, q +� +δi1mσ(1) · · · δikmσ(k)Xj1···jkℓ1···ℓkδj1ℓτ(1) · · · δjklτ(k) +(A3) += +� +j1,...,jk +� +σ,τ∈Sk +Wg +� +σ−1τ, q +� +δi1mσ(1) · · · δikmσ(k)Xj1···jkjτ−1(1)···jτ−1(k) +(A4) += +� +σ,τ∈Sk +Wg +� +σ−1τ, q +�� +P (q) +σ−1 +� +i1···ikm1···mk +tr +� +XP (q) +τ +� +, +(A5) +where, in the final line, we have used that +� +P (q) +σ−1 +� +i1···ikm1···mk += ⟨i1 · · · ik|P (q) +σ−1|m1 · · · mk⟩ +(A6) += ⟨i1 · · · ik|mσ(1) · · · mσ(k)⟩ +(A7) += δi1mσ(1) · · · δikmσ(k) +(A8) +and that +tr +� +XP (q) +τ +� += +� +j1,...,jk +⟨j1 · · · jk|XP (q) +τ +|j1 · · · jk⟩ +(A9) += +� +j1,...,jk +⟨j1 · · · jk|X|jτ −1(1) · · · jτ −1(k)⟩ +(A10) += Xj1···jkjτ−1(1)···jτ−1(k). +(A11) +Thus, +T (k) +U +(X) = +� +σ,τ∈Sk +Wg +� +σ−1τ, q +� +P (q) +σ−1 tr +� +XP (q) +τ +� +(A12) += +� +σ,τ∈Sk +Wg +� +στ −1, q +� +P (q) +σ +tr +� +XP (q) +τ −1 +� +(A13) += +� +σ,τ∈Sk +Wg +� +στ −1, q +� +P (q) +σ +tr +� +X +� +P (q) +τ +�T � +, +(A14) +which coincides with Eq. (19). +In addition, let us confirm that T (k) +U +� +P (q) +ρ +� += P (q) +ρ +[52, 53]. Indeed, +T (k) +U +� +P (q) +ρ +� += +� +σ,τ∈Sk +Wg +� +στ −1, q +� +P (q) +σ +tr +� +P (q) +ρ +� +P (q) +τ +�T � +(A15) += +� +σ,τ∈Sk +Wg +� +στ −1, q +� +P (q) +σ q#(ρτ −1) +(A16) += +� +σ∈Sk +δρσP (q) +σ +(A17) += P (q) +ρ , +(A18) +where, in the third line, we have used the definition of the Weingarten function. +Appendix B: Proofs of Propositions 1 and 2 +Proposition 1. The eigenvalues of Te with e ∈ Sk are non-negative for any k ≥ 2. + +21 +Proposition 2. Te with e ∈ Sk is diagonalizable for any k ≥ 2. +To prove Propositions 1 and 2, we will define matrices X ∈ Rk!×k! and Y ∈ Rk!×k! so that Ck = WXY . We will +then discuss some properties of those three matrices. The proofs themselves will boil down to similarity. +We define the diagonal matrix X ∈ Rk!×k! via +Xσσ = d#(σ). +(B1) +It is evident that X is positive definite. +We define the Gram matrix Y ∈ Rk!×k! via +Yσθ = tr +� +P (D) +σ +� +P (D) +θ +�T � += D#(σθ−1). +(B2) +Y is positive semidefinite because it is a Gram matrix. +The Weingarten matrix Wg +� +στ −1, q +� += +� +G−1� +στ is positive definite because the Gram matrix G [see Eq. (21)] is +positive definite [72]. +We will need the fact that Y W is positive semidefinite. Y W has non-negative eigenvalues because it is a product +of a positive semidefinite (Y ) and a positive definite matrix (W) (see Corollary 7.6.2 of Ref. [73]). Furthermore, Y W +is symmetric: +⟨τ|Y W|θ⟩ = +� +σ∈Sk +Wg +� +στ −1, dD +� +D#(σθ−1) +(B3) += +� +π∈Sk +Wg +� +πθτ −1, dD +� +D#(π) +(B4) += +� +π∈Sk +Wg +� +τθ−1π−1, dD +� +D#(π) +(B5) += +� +π∈Sk +Wg +� +θ−1π−1τ, dD +� +D#(π) +(B6) += +� +ϕ∈Sk +Wg +� +θ−1ϕ, dD +� +D#(τϕ−1) +(B7) += +� +ϕ∈Sk +Wg +� +ϕθ−1, dD +� +D#(ϕτ −1) +(B8) += ⟨θ|Y W|τ⟩ +(B9) +In the third line, we have used that Wg(α, q) = Wg +� +α−1, q +� +, and, in the fourth line, we have used that Wg(α, q) = +Wg +� +βαβ−1, q +� +. Both identities are a result of the Weingarten function being sensitive only to the conjugacy class of +a given permutation. +Proof of Proposition 1. Ck = WXY is similar to XY W. A product of a positive definite (X) and a positive semidef- +inite matrix (Y W), XY W has non-negative eigenvalues. The statement follows by similarity. +Proof of Proposition 2. Ck = WXY is similar to XY W, which is similar to X1/2Y WX1/2. Because Y W is symmetric, +so is X1/2Y WX1/2, which makes the latter diagonalizable. The statement follows. +Appendix C: Proof of Proposition 3 +Proposition 3. Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Te with e ∈ Sk. Then, λ1 = 1 and it is +non-degenerate for any k ≥ 2 if d ≥ 2. +Proof. For what follows, it is convenient to define the transfer matrix Σe with e ∈ Sk via += += +. +(C1) + +22 +Our strategy will be to prove the claimed spectral property for Σe. This is enough because, for any two operators +X and Y for which XY and Y X are well defined, the sets of eigenvalues of XY and Y X are the same (up to zeros +and the multiplicity of the non-zero eigenvalues). By Eqs. (51) and (C1), Tρ and Σρ are related in exactly this way. +As Σe arises from the contraction of the quantum channel underlying R [see Eq. (44)] with the identity permutation, +it can be understood as a generalization of the k-fold twirling operator. In particular, it involves an ”environment” +E of dimension +� +Cd�⊗k that is eventually traced out. As a superoperator (that is, without using the operator-vector +correspondence), it reads [see Eq. (18)] +Σe(·) = +� +dU trE +� +U ⊗k +� k +� +l=1 +|0⟩⟨0|El ⊗ (·)Sl +� +� +U †�⊗k +� +, +(C2) +where the integration is with respect to the Haar measure on the unitary group U(dD), E = �k +l=1 El corresponds +to +� +Cd�⊗k, and S = �k +l=1 Sl corresponds to +� +CD�⊗k. It is apparent that Eq. (C2) represents a convex combination +of quantum channels (note the Stinespring dilation form), and thus the resulting operator is also a valid quantum +channel. This implies that 1 is an eigenvalue of Σe and that there is no eigenvalue of greater modulus [74]. +We now prove that no other eigenvalue of the same modulus exists. To that end, we will show that Σe is a primitive +channel [74], which implies said property. A quantum channel is primitive if and only if the spanning space formed +by products of its Kraus operators, +Km = span +�� m +� +k=1 +Kik +�� +, +(C3) +is equal to the full matrix algebra for some integer m, that is, Km = MDk(C) [75]. We show that this condition is +satisfied for Σe if d ≥ 2. +Indeed, the Haar integral in Eq. (C2), together with the partial trace, can be understood as a (redundant) Kraus +decomposition. Precisely, we can take +{Ki}i = +� +trE +�� k +� +l=1 +|+⟩⟨ψl|El ⊗ ISl +� +U ⊗k +�� +U,ψ +, +(C4) +where U ∈ U(dD), |ψl⟩ ∈ {|0⟩, . . . , |d − 1⟩} is the computational basis of the lth replica of the environment, and +|+⟩ = �d−1 +j=0 |j⟩/ +√ +d. The latter is a choice (instead of |0⟩) made for later convenience. It remains to show that there +exists an integer m = m(k, d, D) such that Km = MDk(C) if d ≥ 2. First of all, note that the above fails for d = 1 +(that is, if the environment E is trivial). This is because span +�� +U ⊗k� +U +� +coincides with the symmetric subspace over +the k subsystems [48]. However, d = 2 is already enough to span the full matrix algebra. +An explicit construction to show this fact amounts to taking U to be a controlled unitary gate, where the control +system is E. In particular, consider +|ψl⟩ = |δlr⟩ +and +U = |0⟩⟨0|E ⊗ IS + +d−1 +� +j=1 +|j⟩⟨j|E ⊗ VS, +(C5) +where r ∈ {1, . . . , k}, and VS is unitary. This results in Kraus operators [see Eq. (C4)] of the form +IS1 ⊗ · · · ⊗ ISr−1 ⊗ V ⊗ ISr+1 · · · ⊗ ISk. +(C6) +Taking (finite) products, as Eq. (C3) dictates, is enough to build a basis of the vector space MDk(C). Note that this +construction requires two control levels (that is, d ≥ 2). +This concludes the proof. +Appendix D: Further Properties of the Transfer Matrix Te +In this appendix, we state and prove statements about the structure of the transfer matrix Te with e ∈ Sk that will +motivate Conjecture 1. We prove the statements in Apps. E, F, and G. +Moving forward, we denote Te with e ∈ Sk by Ck. Each entry +⟨τ|Ck|θ⟩ = +� +σ∈Sk +Wg +� +στ −1, dD +� +d#(σ)D#(σθ−1) +(D1) + +23 +of Ck ∈ Rk!×k! is a sum of k! terms. +While this may sound daunting, Ck exhibits a structure that reduces its +complexity. +As formalized by Proposition 4, the entries ⟨τ|Ck|θ⟩ of Ck exhibit a dependence on the conjugacy class of their +indices. In particular, if we know the entries of the column given by θ ∈ Sk, we also know the entries of the columns +given by permutations in the same conjugacy class as θ. +Proposition 4. For any π ∈ Sk, +⟨τ|Ck|θ⟩ = ⟨πτπ−1|Ck|πθπ−1⟩. +(D2) +Certain entries of Ck vanish, while others are given by the entries of Ck−1. +Proposition 5 captures these two +statements. +Proposition 5. For all θ ∈ Sk with θ(k) = k, +⟨τ|Ck|θ⟩ = δk,τ(k)⟨τ ↓|Ck−1|θ↓⟩, +(D3) +where ρ↓ ∈ Sk−1 is the restriction of ρ ∈ Sk with ρ(k) = k to the permutation on {1, . . . , k − 1}. +To understand the strength of Propositions 4 and 5, let us have a look at C2 and C3, the two most simple transfer +matrices. With bases S2 = {e, (12)} and S3 = {e, (12), (13), (23), (123), (132)}, one finds that +C2 = +� +1 α +0 β +� +and +C3 = +� +� +� +� +� +� +� +1 α α α γ γ +0 β 0 0 δ δ +0 0 β 0 δ δ +0 0 0 β δ δ +0 0 0 0 ε ζ +0 0 0 0 ζ ε +� +� +� +� +� +� +� +, +(D4) +where each Greek letter corresponds to some function of d and D whose exact form is not relevant here. The entries +in the first four columns of C3 are fully determined by those of C2. The entries in the last two columns of C3 do not +arise from those of C2, but the sixth column is a permutation of the fifth. +Note that we are deliberately choosing a basis Sk = {s1, . . . , sk!} that makes the special structure of Ck more +apparent. In particular, we sort permutations so that those with i fixed points come before those with i − 1 fixed +points. We group permutations that have common fixed points and then those that are in the the same conjugacy +class. Given this basis, Propositions 4 and 5 imply that Ck is block triangular with k diagonal blocks. +We denote by C(1) +k +the diagonal block with τ = θ = e ∈ Sk and by C(i) +k +with 2 ≤ i ≤ k the diagonal block +corresponding to τ, θ ∈ Sk with k − i fixed points. The spectrum of Ck is then given by +λ(Ck) = λ +� +C(1) +k +� +∪ λ +� +C(2) +k +� +∪ · · · ∪ λ +� +C(k) +k +� +. +(D5) +It is apparent that C(1) +k +has a single, non-degenerate eigenvalue +µ1 = ⟨e|Ck|e⟩ = 1. +(D6) +C(2) +k +has a single, degenerate eigenvalue +µ2 = ⟨(12)|Ck|(12)⟩ = dD2 − d +d2D2 − 1, +(D7) +which corresponds to the expression of β in Eq. (D4). The degeneracy of µ2 is given by the size of the block, which +is in turn given by the number of transpositions in Sk, +v2 = +�k +2 +� +. +(D8) +By Proposition 3, 1 is the leading eigenvalue of Ck and non-degenerate. We conjecture that µ2 is the subleading +eigenvalue of Ck and that it has degeneracy v2. The statement of Conjecture 1 holds for k ∈ {2, 3, 4}, and we have +numerical evidence suggesting so for k ∈ {5, 6, 7} [63]. + +24 +Conjecture 1. Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Ck. Then, λ2 = µ2 with degeneracy w2 = v2 +for any k ≥ 2. +As formalized by Proposition 6, the statements above hold for any transfer matrix Tρ with ρ ∈ Sk because Tρ is +similar to Ck. +Proposition 6. For any ρ ∈ Sk, +Tρ = QT +ρ CkQρ +with +Qρ = +� +π∈Sk +|ρπ⟩⟨π|. +(D9) +Appendix E: Proof of Proposition 4 +Proposition 4. For any π ∈ Sk, +⟨τ|Ck|θ⟩ = ⟨πτπ−1|Ck|πθπ−1⟩. +(D2) +Proof. It holds that +⟨τ|Ck|θ⟩ = +� +σ∈Sk +Wg +� +στ −1, dD +� +d#(σ)D#(σθ−1) +(E1) += +� +ϕ∈Sk +Wg +� +π−1ϕπτ −1, dD +� +d#(π−1ϕπ)D#(π−1ϕπθ−1) +(E2) += +� +ϕ∈Sk +Wg +� +ϕπτ −1π−1, dD +� +d#(π−1ϕπ)D#(ϕπθ−1π−1) +(E3) += +� +ϕ∈Sk +Wg +� +ϕ +� +πτπ−1�−1, dD +� +d#(ϕ)D +# +� +ϕ(πθπ−1) +−1� +(E4) += ⟨πτπ−1|Ck|πθπ−1⟩, +(E5) +where, in the second line, we have used that the conjugation map is an isomorphism. In the third line, we have used +that Wg(α, q) = Wg +� +βαβ−1, q +� +and that #(α) = # +� +βαβ−1� +. Both identities are a result of the two functions being +sensitive only to the conjugacy class of a given permutation. +Appendix F: Proof of Proposition 5 +Proposition 5. For all θ ∈ Sk with θ(k) = k, +⟨τ|Ck|θ⟩ = δk,τ(k)⟨τ ↓|Ck−1|θ↓⟩, +(D3) +where ρ↓ ∈ Sk−1 is the restriction of ρ ∈ Sk with ρ(k) = k to the permutation on {1, . . . , k − 1}. +To prove Proposition 5, we will need a number of ingredients. The main one will be Proposition 2.2 of Ref. [76], +which we state without proof as Lemma 2. +Lemma 2. For any ρ ∈ Sk, +k−1 +� +i=1 +Wg +� +(ik)π, q +� ++ q Wg(π, q) = δk,π(k) Wg +� +π↓, q +� +, +(F1) +where (ik) denotes the transposition of elements i and k. +To use Lemma 2, we must do two things. First, we must split the sum in Eq. (D1) into certain sums of k terms. +We employ Lemma 3 to achieve this. +Lemma 3. For any α ∈ Sk, there exists a β ∈ Sk with β(k) = k such that either α = β or α = (ik)β with +i ∈ {1, . . . , k − 1}. + +25 +Proof. If α(k) = k, then α = β. Otherwise, k is in exactly one of the disjoint cycles of α. Without loss of generality, +say α = (kia3a4 · · · )(b1b2 · · · ) · · ·. Then, α = (ik)(ia3a4 · · · )(b1b2 · · · ) · · · ≡ (ik)β. +Second, we must ensure that summands in Eq. (D1) with Wg(π, dD) have a factor dD while those with +Wg +� +(ik)π, dD +� +do not. We achieve this with Lemma 5 whose proof employs Lemma 4 of Ref. [77], which we state +without proof as Lemma 4. The lemma also appears as Lemma 1 in Ref. [78]. +Lemma 4. Let α ∈ Sk be a transposition, and β ∈ Sk such that #(β) = u. If the elements exchanged by α are not +in the same cycle of β, then #(αβ) = u − 1. +Lemma 5. Let ϕ, θ ∈ Sk with ϕ(k) = k and θ(k) = k. Then, +1. # +� +ϕθ−1� += # +� +(ik)ϕθ−1� ++ 1 for all i ∈ {1, . . . , k − 1}. +2. # +� +ϕθ−1� +− #(ϕ) = # +� +(ik)ϕθ−1� +− # +� +(ik)ϕ +� +for all i ∈ {1, . . . , k − 1}. +Proof of 1. Let ϕ, θ ∈ Sk with ϕ(k) = k and θ(k) = k. Then, +� +ϕθ−1� +(k) = k. That is, k is in a cycle by itself and +thus not in the same cycle of ϕθ−1 as i. The statement follows with Lemma 4. +Proof of 2. Let ϕ, θ ∈ Sk with ϕ(k) = k and θ(k) = k. Then, i and k are not in the same cycle of neither ϕθ−1 nor +ϕ. By Lemma 4, # +� +ϕθ−1� += # +� +(ik)ϕθ−1� ++ 1 and #(ϕ) = # +� +(ik)ϕ +� ++ 1. Thus, # +� +ϕθ−1� +− #(ϕ) = # +� +(ik)ϕθ−1� ++ +1 − # +� +(ik)ϕ +� +− 1 = +� +(ik)ϕθ−1� +− # +� +(ik)ϕ +� +. +With that, we can prove Proposition 5. +Proof of Proposition 5. By Lemma 3, +⟨τ|Ck|θ⟩ = +� +σ∈Sk +Wg +� +στ −1, dD +� +d#(σ)D#(σθ−1) +(F2) += +� +ϕ∈Sk +ϕ(k)=k +�k−1 +� +i=1 +Wg +� +(ik)ϕτ −1, dD +� +d# +� +(ik)ϕ +� +D#((ik)ϕθ−1) + Wg +� +ϕτ −1, dD +� +d#(ϕ)D#(ϕθ−1) +� +. +(F3) +By Lemma 5, for θ ∈ Sk with θ(k) = k, +⟨τ|Ck|θ⟩ = +� +ϕ∈Sk +ϕ(k)=k +1 +d#(ϕθ−1)−#(ϕ) +�k−1 +� +i=1 +Wg +� +(ik)ϕτ −1, dD +� +(dD)#((ik)ϕθ−1) + Wg +� +ϕτ −1, dD +� +(dD)#(ϕθ−1) +� +(F4) += +� +ϕ∈Sk +ϕ(k)=k +(dD)#(ϕθ−1)−1 +d#(ϕθ−1)−#(ϕ) +�k−1 +� +i=1 +Wg +� +(ik)ϕτ −1, dD +� ++ dD Wg +� +ϕτ −1, dD +� +� +(F5) += +� +ϕ∈Sk +ϕ(k)=k +d#(ϕ)−1D#(ϕθ−1)−1 +�k−1 +� +i=1 +Wg +� +(ik)ϕτ −1, dD +� ++ dD Wg +� +ϕτ −1, dD +� +� +(F6) += +� +ϕ∈Sk +ϕ(k)=k +d#(ϕ↓)D +# +� +(ϕθ−1] +↓��k−1 +� +i=1 +Wg +� +(ik)ϕτ −1, dD +� ++ dD Wg +� +ϕτ −1, dD +� +� +, +(F7) +where, in the final line, we have used that # +� +α↓� += #(α) − 1. +By Lemma 2, +⟨τ|Ck|θ⟩ = +� +ϕ∈Sk +ϕ(k)=k +δk,(ϕτ −1)(k) Wg +�� +ϕτ −1�↓, dD +� +d#(ϕ↓)D +# +� +(ϕθ−1) +↓� +(F8) += δk,τ(k) +� +ϕ∈Sk +ϕ(k)=k +Wg +�� +ϕτ −1�↓, dD +� +d#(ϕ↓)D +# +� +(ϕθ−1) +↓� +(F9) += δk,τ(k)⟨τ ↓|Ck−1|θ↓⟩, +(F10) + +26 +where, in the second line, we have used that +� +ϕτ −1� +(k) = k if and only if τ(k) = k because ϕ(k) = k. +This concludes the proof. +Appendix G: Proof of Proposition 6 +Proposition 6. For any ρ ∈ Sk, +Tρ = QT +ρ CkQρ +with +Qρ = +� +π∈Sk +|ρπ⟩⟨π|. +(D9) +Proof. It holds that +⟨τ|Tρ|θ⟩ = +� +σ∈Sk +Wg +� +στ −1, dD +� +d#(σρ)D#(σθ−1) +(G1) += +� +σ∈Sk +Wg +� +ρστ −1ρ−1, dD +� +d#(ρσ)D#(ρσθ−1ρ−1) +(G2) += +� +σ∈Sk +Wg +� +ρσ(ρτ)−1, dD +� +d#(ρσ)D#[ρσ(ρθ)−1] +(G3) += +� +π∈Sk +Wg +� +π(ρτ)−1, dD +� +d#(π)D#[π(ρθ)−1] +(G4) += ⟨ρτ|Ck|ρθ⟩, +(G5) +where, in the second line, we have used that Wg(α, q) = Wg +� +βαβ−1, q +� +and that #(α) = # +� +βαβ−1� +. Both identities +are a result of the two functions being sensitive only to the conjugacy class of a given permutation. In the fourth line, +we have used that the left-multiplication map is an isomorphism. +Appendix H: Proof of Result 2 +Result 2. The average of N(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in +Fig. 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq. (34). +Proof. We split the proof into four steps, following the structure of the proof of Result 1. +Step 1. We rewrite EN(A : B) in terms of expressions of the form of Eq. (40). With the Hilbert-Schmidt inner +product, +EN(A : B) = E tr +� +ϱ2 +AB +� ++ E tr +� +ϱ2 +A +� +tr +� +ϱ2 +B +� +− 2E tr[ϱAB(ϱA ⊗ ϱB)]. +(H1) +It is then easy to confirm that +EN(A : B) = tr +��� +P (d) +e +�⊗c +⊗ +� +P (d) +(12) +�⊗a +⊗ +� +P (d) +e +�⊗r +⊗ +� +P (d) +(12) +�⊗b +⊗ +� +P (d) +e +�⊗(f−1) +⊗ P (dD) +e +� +E|ψ⟩⟨ψ|⊗4 +� ++ tr +��� +P (d) +e +�⊗c +⊗ +� +P (d) +(12) +�⊗a +⊗ +� +P (d) +e +�⊗r +⊗ +� +P (d) +(34) +�⊗b +⊗ +� +P (d) +e +�⊗(f−1) +⊗ P (dD) +e +� +E|ψ⟩⟨ψ|⊗4 +� +− 2 tr +��� +P (d) +e +�⊗c +⊗ +� +P (d) +(12) +�⊗a +⊗ +� +P (d) +e +�⊗r +⊗ +� +P (d) +(13) +�⊗b +⊗ +� +P (d) +e +�⊗(f−1) +⊗ P (dD) +e +� +E|ψ⟩⟨ψ|⊗4 +� +. (H2) +Step 2. We express EN(A : B) in terms of the transfer matrices defined in Sec. IV A. Given the previous step, it is +easy to confirm that +EN(A : B) = ⟨I4|T c +e T a +(12)T r +e T b +(12)|F4⟩ + ⟨I4|T c +e T a +(34)T r +e T b +(12)|F4⟩ − 2⟨I4|T c +e T a +(12)T r +e T b +(13)|F4⟩ +(H3) += ⟨I4|T c +e T a +(12)T r +e +� +T b +(12) + T b +(34) − 2T b +(13) +� +|F4⟩ +(H4) +≡ ⟨I4|T c +e AT r +e B|F4⟩, +(H5) +where we have defined +A = T a +(12) +and +B = T b +(12) + T b +(34) − 2T b +(13). +(H6) + +27 +Step 3. We expand EN(A : B) in terms of the spectrum of Te with e ∈ S4 [see Eq. (58)]. Let λ1 > λ2 > · · · ≥ 0 +denote the distinct eigenvalues of Te. It holds [63] that +λ1 = 1 +and +λ2 = dD2 − d +d2D2 − 1. +(H7) +The former is non-degenerate, while the degeneracy of the latter is given by the number of transposition in S4 [63], +w2 = +�4 +2 +� += 6. +(H8) +Expanding T c +e and taking the limit c → ∞ yields +EN(A : B) = ⟨L1|AT r +e B|F4⟩, +(H9) +where we have used that ⟨I4|R1⟩ = 1. After expanding T r +e and using that |F4⟩ = |R1⟩, we have +EN(A : B) = ⟨L1|A|R1⟩⟨L1|B|R1⟩ + λr +2 +w2 +� +µ=1 +⟨L1|A|R(µ) +2 ⟩⟨L(µ) +2 |B|R1⟩ + O(λr +3) +(H10) += λr +2 +w2 +� +µ=1 +⟨L1|A|R(µ) +2 ⟩⟨L(µ) +2 |B|R1⟩ + O(λr +3), +(H11) +where, in the second line, we have used that +⟨L1|B|R1⟩ = 0, +(H12) +which we prove in the following. +In particular, we prove that ⟨L1|T b +t |R1⟩ does not depend on the two elements the transposition t ∈ S4 acts upon. +We need some understanding of T b +t as well as the eigenvectors ⟨L1| and |R1⟩ of Te. For the former, we make use of +the fact that T b +ρ with ρ ∈ S4 is similar to to T b +e with e ∈ S4, +T b +ρ = +� +π,ϕ∈S4 +|π⟩⟨ρπ|Cb +4|ρϕ⟩⟨ϕ|. +(H13) +With +α = d2D − D +d2D2 − 1, +β = dD2 − d +d2D2 − 1, +f(u) = +u−1 +� +i=0 +αβi +and +g(u) = βu, +(H14) + +28 +we have +� +Te|si⟩ +� +1≤i≤7 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 α α α α α α +0 β 0 0 0 0 0 +0 0 β 0 0 0 0 +0 0 0 β 0 0 0 +0 0 0 0 β 0 0 +0 0 0 0 0 β 0 +0 0 0 0 0 0 β +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +0 0 0 0 0 0 0 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +and +� +T b +e |si⟩ +� +1≤i≤7 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 f(b) f(b) f(b) f(b) f(b) f(b) +0 g(b) +0 +0 +0 +0 +0 +0 +0 +g(b) +0 +0 +0 +0 +0 +0 +0 +g(b) +0 +0 +0 +0 +0 +0 +0 +g(b) +0 +0 +0 +0 +0 +0 +0 +g(b) +0 +0 +0 +0 +0 +0 +0 +g(b) +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +. (H15) +For getting some understanding of ⟨L1|, we make use of the fact that ⟨L(µ) +i +|R(ν) +j ⟩ = δijδµν. It is easy to confirm that +|R1⟩ = |s1⟩ +and +|R(µ) +2 ⟩ = +α +β − 1|s1⟩ + |sµ+1⟩, +(H16) +which implies that +� +⟨L1|si⟩ +� +1≤i≤7 = +� +1 +α +β − 1 +α +β − 1 +α +β − 1 +α +β − 1 +α +β − 1 +α +β − 1 +� +. +(H17) +For any transposition t ∈ S4, it thus holds that +⟨L1|T b +t |R1⟩ = +� +π,ϕ∈S4 +⟨L1|π⟩⟨tπ|Cb +4|tϕ⟩⟨ϕ|R1⟩ +(H18) += +� +π,ϕ∈S4 +⟨L1|π⟩⟨tπ|Cb +4|tϕ⟩δs1ϕ +(H19) += +� +π∈S4 +⟨L1|π⟩⟨tπ|Cb +4|t⟩ +(H20) += +� +π∈S4 +f(b)⟨L1|π⟩⟨tπ|s1⟩ + +� +π∈S4 +g(b)⟨L1|π⟩⟨tπ|t⟩ +(H21) += +� +π∈S4 +f(b)⟨L1|π⟩δtπ + +� +π∈S4 +g(b)⟨L1|π⟩δs1π +(H22) += f(b)⟨L1|t⟩ + g(b)⟨L1|s1⟩ +(H23) += +α +β − 1f(b) + g(b), +(H24) +which is independent of the two elements the transposition t ∈ S4 acts upon. Thus, +⟨L1|B|R1⟩ = ⟨L1| +� +T b +(12) + T b +(34) − 2T b +(13) +� +|R1⟩ = 0. +(H25) + +29 +Step 4. Finally, we can write EN(A : B) in the form of Definition 1. That is, +EN(A : B) ≡ K exp +� +−r +ξ +� ++ O +� +exp +� +− r +χ +�� +, +(H26) +where +K = +w2 +� +µ=1 +⟨L1|A|R(µ) +2 ⟩⟨L(µ) +2 |B|R1⟩ +(H27) +and +ξ = − +1 +log(λ2) = − +� +log +� dD2 − d +d2D2 − 1 +��−1 += ξ1D > χ. +(H28) +This concludes the proof. +Appendix I: Proof of Result 3 and Corollary 4 +Result 3. If Conjecture 1 holds, the average of I(A : B) with respect to the random MPS ensemble and subsystems +A and B as sketched in Fig. 2 (a) decays exponentially as specified in Definition 1 with the average correlation length +ξ1D defined in Eq. (34). +Corollary 4. If Conjecture 1 holds, for any integer value of α ≥ 1, the average of Iα(A : B) with respect to the random +MPS ensemble and subsystems A and B as sketched in Fig. 2 (a) decays exponentially as specified in Definition 1 +with the average correlation length ξ1D defined in Eq. (34). +In this appendix, we prove Result 3 and Corollary 4. The proof of the latter will follow directly from the proof of +the former. +Proof of Result 3. We split the proof into four steps, following the structure of the proof of Result 1. Because I(A : B) +and I2(A : B) are related, the steps are overall very similar. As is usual in the context of the replica trick [79], we +will interchange the order of some limits without rigorous justification. +Step 1. We rewrite EI(A : B) in terms of expressions of the form of Eq. (40). To that end, we make use of Eqs. (88) +and (89). With those, +EI(A : B) = lim +α→1 lim +v→0 +1 +vα − v +� +log +� +E tr(ϱα +AB)v� +− log +� +E tr(ϱα +A)v� +− log +� +E tr(ϱα +B)v�� +. +(I1) +E tr(ϱα +A)v, E tr(ϱα +A)v, and E tr(ϱα +AB)v can be written in the desired form. +Let us define x ∈ Sα to be the cyclic +permutation so that x(i) = i + 1 modulo α and +xw = +� +α(w − 1) + 1, α(w − 1) + 2, . . . , αw +� +∈ Svα +(I2) +with w ∈ {1, . . . , v}. Then, for example, +E tr(ϱα +AB)v = tr +��� +P (d) +e +�⊗c +⊗ +� +P (d) +x +�⊗a +⊗ +� +P (d) +e +�⊗r +⊗ +� +P (d) +x +�⊗b +⊗ +� +P (d) +e +�⊗(f−1) +⊗ P (dD) +e +� +E|ψ⟩⟨ψ|⊗α +�v +(I3) += tr +��� +P (d) +e +�⊗c +⊗ +� +P (d) +x1···xv +�⊗a +⊗ +� +P (d) +e +�⊗r +⊗ +� +P (d) +x1···xv +�⊗b +⊗ +� +P (d) +e +�⊗(f−1) +⊗ P (dD) +e +� +E|ψ⟩⟨ψ|⊗vα +� +. (I4) +Step 2. We express EI(A : B) in terms of the transfer matrices defined in Sec. IV A. Given the previous step, it is +easy to confirm that +EI(A : B) = lim +α→1 lim +v→0 +1 +vα − v +� +log +� +⟨Ivα|T c +e T a +x1···xvT r +e T b +x1···xv|Fvα⟩ +� +− log +� +⟨Ivα|T c +e T a +x1···xv|Fvα⟩ +� +− log +� +⟨Ivα|T c+a+r +e +T b +x1···xv|Fvα⟩ +�� +(I5) +≡ lim +α→1 lim +v→0 +1 +vα − v +� +log(⟨Ivα|T c +e AT r +e B|Fvα⟩) − log(⟨Ivα|T c +e A|Fvα⟩) − log +� +⟨Ivα|T c+a+r +e +B|Fvα⟩ +�� +(I6) +where we have defined +A = T a +x1···xv +and +B = T b +x1···xv. +(I7) + +30 +Step 3. We expand EI(A : B) in terms of the spectrum of Te with e ∈ Svα [see Eq. (58)]. At this point, we assume +Conjecture 1 to hold. That is, for any vα ≥ 2, we assume that +λ2 = dD2 − d +d2D2 − 1. +(I8) +with degeneracy +w2 = +�k +2 +� +. +(I9) +Expanding T c +e and taking the limit c → ∞ yields +EI(A : B) = lim +α→1 lim +v→0 +1 +vα − v [log(⟨L1|AT r +e B|Fvα⟩) − log(⟨L1|A|Fvα⟩) − log(⟨L1|B|Fvα⟩)], +(I10) +where we have used that ⟨Ivα|R1⟩ = 1. After expanding T r +e and using that |Fvα⟩ = |R1⟩, we have +EI(A : B) = lim +α→1 lim +v→0 +1 +vα − v +� +log +� +⟨L1|A|R1⟩⟨L1|B|R1⟩ + λr +2 +w2 +� +µ=1 +⟨L1|A|R(µ) +2 ⟩⟨L(µ) +2 |B|R1⟩ + O(λr +3) +� +− log(⟨L1|A|R1⟩) − log(⟨L1|B|R1⟩) +� +. +(I11) +Step 4. Finally, we can write EI(A : B) in the form of Definition 1. With Λ = max +�� +λ2r +2 , λr +3 +�� +, +EI(A : B) = lim +α→1 lim +v→0 +1 +vα − v log +� +1 + λr +2 +w2 +� +µ=1 +⟨L1|A|R(µ) +2 ⟩⟨L(µ) +2 |B|R1⟩ +⟨L1|A|R1⟩⟨L1|B|R1⟩ ++ O(λr +3) +� +(I12) += lim +α→1 lim +v→0 +1 +vα − v +� +λr +2 +w2 +� +µ=1 +⟨L1|A|R(µ) +2 ⟩⟨L(µ) +2 |B|R1⟩ +⟨L1|A|R1⟩⟨L1|B|R1⟩ ++ O(Λ) +� +(I13) +≡ lim +α→1 lim +v→0 +1 +vα − v +� +�K(vα) exp +� +−r +ξ +� ++ O +� +exp +� +− r +χ +��� +, +(I14) +where +�K(vα) = +w2 +� +µ=1 +⟨L1|A|R(µ) +2 ⟩⟨L(µ) +2 |B|R1⟩ +⟨L1|A|R1⟩⟨L1|B|R1⟩ +(I15) +and +ξ = − +1 +log(λ2) = − +� +log +� dD2 − d +d2D2 − 1 +��−1 += ξ1D > χ. +(I16) +As ξ is independent of vα, it cannot be affected by the replica limits. �K(vα) will converge to some K that is guaranteed +to be independent of r. +This concludes the proof. +Proof of Corollary 4. Using Eq. (89), it holds that +EIα(A : B) = lim +v→0 +1 +vα − v +� +log +� +E tr(ϱα +AB)v� +− log +� +E tr(ϱα +A)v� +− log +� +E tr(ϱα +B)v�� +. +(I17) +Thus, the proof is identical to that of Result 3 without the limit α → 1. The statement follows. + +31 +Appendix J: Result 3 with c = 0 +In this appendix, we prove a version of Result 3 with c = 0. Steps 1 and 2 of this proof are identical to Steps 1 and +2 of the proof of Result 3 with c = 0. That is, at the end of Step 2, we have +EI(A : B) = lim +α→1 lim +v→0 +1 +vα − v +� +log(⟨Ivα|ACr +vαB|Fvα⟩) − log(⟨Ivα|A|Fvα⟩) − log +� +⟨Ivα|Ca+r +vα B|Fvα⟩ +�� +. +(J1) +We start the proof at Step 3. +Step 3. We expand EI(A : B) in terms of the spectrum of Te with e ∈ Svα [see Eq. (58)]. We assume Conjecture 1 +to hold. Expanding T r +e and using that ⟨Ivα|R1⟩ = 1 yields +EI(A : B) = lim +α→1 lim +v→0 +1 +vα − v +� +log +� +⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ + λr +2 +w2 +� +µ=1 +⟨Ivα|A|R(µ) +2 ⟩⟨L(µ) +2 |B|Fvα⟩ + O(λr +3) +� +− log(⟨Ivα|A|Fvα⟩) +− log +� +⟨L1|B|Fvα⟩ + λa+r +2 +w2 +� +µ=1 +⟨Ivα|R(µ) +2 ⟩⟨L(µ) +2 |B|Fvα⟩ + O(λr +3) +�� +. +(J2) +Step 4. We write EI(A : B) in the form of Definition 1. With Λ = max +�� +λ2r +2 , λr +3 +�� +, +EI(A : B) = lim +α→1 lim +v→0 +1 +vα − v +� +log +� +1 + λr +2 +w2 +� +µ=1 +⟨Ivα|A|R(µ) +2 ⟩⟨L(µ) +2 |B|Fvα⟩ +⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ ++ O(λr +3) +� +− log +� +1 + λa+r +2 +w2 +� +µ=1 +⟨Ivα|R(µ) +2 ⟩⟨L(µ) +2 |B|Fvα⟩ +⟨L1|B|Fvα⟩ + O(λr +3) +�� +(J3) += lim +α→1 lim +v→0 +1 +vα − v +� +λr +2 +w2 +� +µ=1 +⟨Ivα|A|R(µ) +2 ⟩⟨L(µ) +2 |B|Fvα⟩ +⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ ++ λa+r +2 +w2 +� +µ=1 +⟨Ivα|R(µ) +2 ⟩⟨L(µ) +2 |B|Fvα⟩ +⟨L1|B|Fvα⟩ + O(Λ) +� +(J4) += lim +α→1 lim +v→0 +1 +vα − v +� +λr +2 +w2 +� +µ=1 +� +⟨Ivα|A|R(µ) +2 ⟩⟨L(µ) +2 |B|Fvα⟩ +⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ ++ λa +2⟨Ivα|R(µ) +2 ⟩⟨L(µ) +2 |B|Fvα⟩ +⟨L1|B|Fvα⟩ +� ++ O(Λ) +� +(J5) +≡ lim +α→1 lim +v→0 +1 +vα − v +� +� +K′(vα) exp +� +−r +ξ +� ++ O +� +exp +� +− r +χ +��� +, +(J6) +where +� +K′(vα) = +w2 +� +µ=1 +� +⟨Ivα|A|R(µ) +2 ⟩⟨L(µ) +2 |B|Fvα⟩ +⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ ++ λa +2⟨Ivα|R(µ) +2 ⟩⟨L(µ) +2 |B|Fvα⟩ +⟨L1|B|Fvα⟩ +� +(J7) +and +ξ = − +1 +log(λ2) = − +� +log +� dD2 − d +d2D2 − 1 +��−1 +ξ1D > χ. +(J8) +Again, � +K′(vα) will converge to some K′ that is guaranteed to be independent of r. While K′ is different from K in +general, the correlation length ξ is independent of c. +Appendix K: Numerical Analysis +In this appendix, we briefly review our numerical analysis of the von Neumann mutual information I(A : B) in +Sec. IV E. We fix d and D, and we set a = b = 1 and r = 5. (i) We generate a + r + b + 1 Haar-random unitary +matrices of U(dD) to define |ψ⟩. This definition makes the assumption that there are no sites before subsystem A +(that is, c = 0). We discuss in App. J why this does not affect the average correlation length. By setting f = 1, we +furthermore use the fact that the sites after subsystem B do not play a role as a result of the sequential generation +[see Eq. (55)]. (ii) We compute I(A : B) with respect to |ψ⟩. (iii) We repeat steps (i) and (ii) 10 000 times to compute +the average of I(A : B). (iv) We repeat steps (i) through (iii) for r ∈ {7, 9, 11, 13, 15}, plot the averages of I(A : B) +against r, and fit the data to extract the average correlation length. (v) To obtain Fig. 3, we repeat steps (i) through +(iv) for different d and D. + +32 +Appendix L: Transfer Matrices in Two Dimensions +This appendix expands on Sec. V A. We will state the definitions of the boundary tensors and prove Eq. (99). +From the main text, recall that we define V (i,j) = U (i,j) ⊗ U (i,j). By computing the k-fold twirl, we obtain the +building block += +� +dU += +. +(L1) +With that, we have +E|ψ⟩⟨ψ|⊗k = += +, +(L2) +where, in the final step, we have cut permutation-valued (green) legs instead of bond (blues) ones. S is the tensor +stated in Eq. (96). S′ and S′′ reflect the different boundary conditions. Instead of first stating the tensors with bond +(blue) legs, let us immediately state those with permutation-valued (green) legs. +The tensors at the top boundary are given via += += +(L3) += +� +σ∈Sk +Wg +� +στ −1, dD2� +(dD)#(σρ)D#(σθ−1), +(L4) +and those at the right boundary are given via += += +(L5) += +� +σ∈Sk +Wg +� +στ −1, dD2� +(dD)#(σρ)D#(σν−1). +(L6) + +33 +We will always contract S′′ with P (d) +e +. The tensor in the top-right corner thus plays the same role as the final vector +|Fk⟩ = e1 ∈ Rk! does in one dimension. In fact, Te = |Fk⟩⟨Fk|, += += +≡ +(L7) += +� +σ∈Sk +Wg +� +στ −1, dD2�� +dD2�#(σe) = δeτ, +(L8) +where we have defined += +. +(L9) +If a tensor corresponding to e ∈ Sk is contracted with |Fk⟩, it factorizes. For tensors at the top boundary, we have += += += +(L10) += +� +σ∈Sk +Wg +� +στ −1, dD2�� +dD2�#(σe) = δeτ, +(L11) +for those at the right boundary, we have += += += +(L12) += +� +σ∈Sk +Wg +� +στ −1, dD2�� +dD2�#(σe) = δeτ, +(L13) +and for those in the bulk, we have += += += +(L14) += +� +σ∈Sk +Wg +� +στ −1, dD2�� +dD2�#(σe) = δeτ. +(L15) + +34 +The identities above lead to Eq. (99): += += += +(L16) += += +. +(L17) +Appendix M: Spectrum of the Transfer Matrix Te with e ∈ S2 +In this appendix, we state and prove two lemmas concerning the spectrum of Te with e ∈ S2. +Let us start with some preliminaries. We define +|0⟩ = +� +1 +0 +� +, +|1⟩ = +� +0 +1 +� +, +and +|+⟩ = +� +1 +1 +� +, +(M1) +and map the contraction of tensors defining Te with e ∈ S2 to a multiplication of matrices: += += +, +(M2) +where += +� +� +� +1 α α γ +0 0 0 0 +0 0 0 0 +0 β β δ +� +� +� +(M3) +with +α = d2D3 − D +d2D4 − 1 , +β = dD3 − dD +d2D4 − 1 , +γ = d2D2 − D2 +d2D4 − 1 , +and +δ = dD2 − d +d2D4 − 1. +(M4) +Note that += 0 +if +i ̸= j, +(M5) + +35 +and +N = += +� +1 α +0 β +� +(M6) +is equal to Te with e ∈ S2 for d → dD. With that, it easy to check that += += ⟨o|N|i⟩⟨i|. +(M7) +We also introduce an analytical notation. We define +Mj = I⊗(j−1) ⊗ M ⊗ I⊗(h−j). +(M8) +Te with e ∈ S2 is then given by +Te = +� +I⊗h ⊗ ⟨0| +�� +Mh · · · M1 +�� +|+⟩ ⊗ I⊗h� +. +(M9) +With i1, . . . , ih ∈ {0, 1} and o1, . . . , oh ∈ {0, 1}, the entries of Te with e ∈ S2 are given by +� +⟨o1, . . . , oh| ⊗ ⟨0| +�� +Mh · · · M1 +�� +|+⟩ ⊗ |i1, . . . , ih⟩ +� +. +(M10) +Our first lemma states that Te with e ∈ S2 is block triangular, where our definition of blocks arises from the indexing +of rows and columns in base 2. In particular, with 2 ≤ j ≤ h, the jth diagonal block of Te, which we denote by +T (j) +e +∈ R2j−1×2j−1, has fixed indices ih = · · · ij+1 = oh = · · · = oj+1 = 0 and ij = oj = 1. Using Eq. (M14), it is given +by +T (j) +e += += +. +(M11) +The first diagonal block, which we denote by T (1) +e +∈ R2×2, is given by +T (1) +e += += += +� +1 α +0 β +� +. +(M12) +In particular, we will prove that a block of Te is zero if its defining row digit oj is higher than its defining column +digit ij, which implies that Te is upper block triangular. As the proof relies exclusively on Eq. (M7), Te inherits its +upper block triangularity from the upper triangularity of N. +Lemma 6. Te with e ∈ S2 is upper block triangular because +� +I⊗(j−1) ⊗ ⟨0| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| +�� +Mh · · · M1 +�� +|+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ ⊗ |0⟩⊗(h−j)� += 0. +(M13) + +36 +Proof. From += |0⟩ +(M14) +and += 0, +(M15) +it follows that += += 0, +(M16) +which concludes the proof. +In our second lemma, we utilize the block triangularity of Te with e ∈ S2 to make a direct statement about its two +leading eigenvalues. +Lemma 7. Let |λ1| > |λ2| > · · · ≥ 0 denote the distinct eigenvalues of Te with e ∈ S2. Then, for any h, λ1 = 1 and +λ2 = β. Furthermore, λ1 and λ2 are non-degenerate. +Proof. The spectrum of Te with e ∈ S2 is given by the union of the spectra of its diagonal blocks. It is evident that the +first block T (1) +e +has eigenvalues 1 and β. In the following, we show that any other diagonal block T (j) +e +with 2 ≤ j ≤ h +can be written as a product of β and a strictly substochastic matrix, implying that its eigenvalues are strictly smaller +than β. We structure the proof in steps. +Step 1. We show that any diagonal block T (j) +e +with 2 ≤ j ≤ h can be written as β times a matrix. From += β|1⟩, +(M17) +it follows that +T (j) +e += += β +. +(M18) +Step 2. We argue that the matrix +(M19) + +37 +is strictly column substochastic. It holds that += +� +� +� +1 α α γ +0 0 0 0 +0 0 0 0 +0 β β δ +� +� +� +(M20) +is column substochastic. While the first column of M evidently sums to 1, the sums of the other columns are strictly +bounded by 1. As a result, Mj−1 · · · M1 is column substochastic. The boundary condition |1⟩ does not affect this +because it specifies a subset of columns of the matrix Mj−1 · · · M1. In fact, it imposes strict substochasticity because +this subset does not include the only column summing to 1. Also the boundary condition ⟨+| does not affect the +substochasticity. The boundary condition means that Eq. (M19) is a sum of matrices. Each of these matrices comprises +a disjoint subset of rows of the matrix Mj−1 · · · M1. Because Mj−1 · · · M1, the matrix comprising the whole set of +rows, is column substochastic, so is the sum of the matrices comprising the disjoint subsets. +Step 3. 1 and β are the only eigenvalues of T (1) +e +. They are non-degenerate. Because the matrix in Eq. (M20) is +strictly column substochastic, the eigenvalues of any diagonal block T (j) +e +with 2 ≤ j ≤ h are strictly smaller than β. +The statement follows. +As a preparation for App. N, we provide the proofs of Lemmas 6 and 7 in analytical notation. +Proof of Lemma 6 in analytical notation. From +� +⟨0| ⊗ ⟨0| +� +M +� +I ⊗ |0⟩ +� += |0⟩ +(M21) +and +� +⟨0| ⊗ ⟨0| +� +M +� +I ⊗ |1⟩ +� += 0, +(M22) +it follows that +� +I⊗(j−1) ⊗ ⟨0| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| +�� +Mh · · · M1 +�� +|+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ ⊗ |0⟩⊗(h−j)� += +� +I⊗(j−1) ⊗ ⟨0| ⊗ ⟨0| +�� +Mj · · · M1 +�� +|+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ +� +(M23) += 0, +(M24) +which concludes the proof. +Proof of Lemma 7 in analytical notation. As in the version with graphical notation, we structure the proof in steps, +without repeating the details. +Step 1. From +� +⟨1| ⊗ ⟨0| +� +M +� +I ⊗ |0⟩ +� += β|1⟩, +(M25) +it follows that +T (j) +e += +� +I⊗(j−1) ⊗ ⟨1| ⊗ ⟨0| +�� +Mj · · · M1 +�� +|+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ +� +(M26) += β +� +I⊗(j−1) ⊗ ⟨1| +�� +Mj−1 · · · M1 +�� +|+⟩ ⊗ I⊗(j−1)� +. +(M27) +Step 2. It holds that +� +I⊗(j−1) ⊗ ⟨1| +�� +Mj−1 · · · M1 +�� +|+⟩ ⊗ I⊗(j−1)� +(M28) +is strictly column substochastic. +Step 3. 1 and β are the only eigenvalues of T (1) +e +. They are non-degenerate. Because the matrix in Eq. (M28) is +strictly column substochastic, the eigenvalues of any diagonal block T (j) +e +with 2 ≤ j ≤ h are strictly smaller than β. +The statement follows. + +38 +Appendix N: Spectrum of the Transfer Matrix Te with e ∈ S4 +In this appendix, we state and prove two lemmas concerning the spectrum of Te with e ∈ S4. Using the same +notation as in App. M, we will draw on results from that appendix. +Te with e ∈ S4 is defined by Eq. (M2), now with M ∈ R576×576. As in App. M, it holds that +� +⟨o| ⊗ ⟨0| +� +M +� +I ⊗ |i⟩ +� += ⟨o|N|i⟩⟨i|, +(N1) +where +N = +� +⟨+| ⊗ I +� +M +� +I ⊗ |0⟩ +� += +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 α α α α α α γ γ γ γ γ γ γ γ η η η η η η ρ ρ ρ +0 β 0 0 0 0 0 δ δ δ δ 0 0 0 0 θ θ +ι θ +ι θ σ τ +τ +0 0 β 0 0 0 0 δ δ 0 0 δ δ 0 0 ι θ θ θ θ +ι τ +σ τ +0 0 0 β 0 0 0 0 0 δ δ δ δ 0 0 θ +ι θ +ι θ θ τ +τ +σ +0 0 0 0 β 0 0 δ δ 0 0 0 0 δ δ θ +ι θ +ι θ θ τ +τ +σ +0 0 0 0 0 β 0 0 0 δ δ 0 0 δ δ +ι θ θ θ θ +ι τ +σ τ +0 0 0 0 0 0 β 0 0 0 0 δ δ δ δ θ θ +ι θ +ι θ σ τ +τ +0 0 0 0 0 0 0 ε ζ 0 0 0 0 0 0 κ κ λ λ κ λ υ υ υ +0 0 0 0 0 0 0 ζ ε 0 0 0 0 0 0 λ λ κ κ λ κ υ υ υ +0 0 0 0 0 0 0 0 0 ε ζ 0 0 0 0 κ κ κ λ λ λ υ υ υ +0 0 0 0 0 0 0 0 0 ζ ε 0 0 0 0 λ λ λ κ κ κ υ υ υ +0 0 0 0 0 0 0 0 0 0 0 ε ζ 0 0 κ λ κ κ λ λ υ υ υ +0 0 0 0 0 0 0 0 0 0 0 ζ ε 0 0 λ κ λ λ κ κ υ υ υ +0 0 0 0 0 0 0 0 0 0 0 0 0 ε ζ κ λ λ κ κ λ υ υ υ +0 0 0 0 0 0 0 0 0 0 0 0 0 ζ ε λ κ κ λ λ κ υ υ υ +0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 µ ν ν ν ν ξ τ ϕ τ +0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ν µ ν ξ ν ν τ +τ ϕ +0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ν ν µ ν ξ ν ϕ τ +τ +0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ν ξ ν µ ν ν τ +τ ϕ +0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ν ν ξ ν µ ν ϕ τ +τ +0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ξ ν ν ν ν µ τ ϕ τ +0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o π o π o χ ψ ψ +0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 π o o o o π ψ χ ψ +0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o π o π o o ψ ψ χ +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(N2) +is equal to Te with e ∈ S4 for d → dD. +As in App. M, our first lemma states that Te with e ∈ S4 is block triangular. The definition of blocks now arises +from the indexing of rows and columns in base 24. In particular, any diagonal block (but the first) is defined by +ih = · · · = ij+1 = oh = · · · = oj+1 = 0 and ij = oj ̸= 0. There are four classes of diagonal blocks: +• The first diagonal block is in its own class. It is given by +� +I ⊗ ⟨0|⊗(h−1) ⊗ ⟨0| +�� +Mh · · · M1 +�� +|+⟩ ⊗ I ⊗ |0⟩⊗(h−1)� += +� +I ⊗ ⟨0| +� +M +� +|+⟩ ⊗ I +� += N. +(N3) +• The second class of diagonal blocks corresponds to transpositions. ij and oj correspond to the same transposition. +There are six subblocks in this class because there are six different transpositions in S4. +• The third class of diagonal blocks corresponds to permutations with a single fixed point. ij and oj correspond +to any of the two permutations with the same single fixed point. There are four subblocks in this class because +there are four different choices of a single fixed point. +• The fourth class of diagonal blocks corresponds to permutations with no fixed point. ij and oj correspond to +any of the nine permutations with no fixed point. +In particular, we will prove that a block of Te is zero if its defining row digit oj is higher than its defining column +digit ij stand in a certain relation to each other. As the proof relies exclusively on Eq. (N1), Te again inherits its +upper block triangularity from the upper triangularity of N. +Lemma 8. Te with e ∈ S4 is upper block triangular. + +39 +Proof. From Eqs. (N1), it follows that +� +I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| +�� +Mh · · · M1 +�� +|+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� += 0 +(N4) +if +• ij corresponds to the trivial permutation and oj does not, +• ij corresponds to a transposition and oj corresponds to a different transposition or a permutation with one or +no fixed point, +• ij corresponds to a permutation with a single fixed point and oj corresponds to a permutation with a different +single fixed point or no fixed point, +which concludes the proof. +In our second lemma, we utilize the block triangularity of Te with e ∈ S4 to make direct a statement about its two +leading eigenvalues. +Lemma 9. Let |λ1| > |λ2| > · · · ≥ 0 denote the distinct eigenvalues of Te with e ∈ S4. Then, for any h, λ1 = 1 and +λ2 = β. Furthermore, λ1 is non-degenerate, and λ2 has a degeneracy of six. +Proof. As is the case for Te with e ∈ S2, the spectrum of Te with e ∈ S4 is given by the union of the spectra of its +diagonal blocks. The two leading eigenvalues of the first diagonal block N are 1 and β. In the following, we show that +any other diagonal block can be written as a product of β and a matrix whose spectral radius is strictly bounded by +1, implying that its eigenvalues are strictly smaller than β. We again structure the proof in steps. +Step 1. We show that any diagonal block but the first can be written as a product of beta and a matrix. From +Eq. (N1), it follows that, +• if ij and oj correspond to the same transposition, +� +I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| +�� +Mh · · · M1 +�� +|+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� += +� +I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0| +�� +Mj · · · M1 +�� +|+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ +� +(N5) += β +� +I⊗(j−1) ⊗ ⟨oj| +�� +Mj−1 · · · M1 +�� +|+⟩ ⊗ I⊗(j−1)� +, +(N6) +• if ij and oj correspond to any two permutations with the same single fixed point, +� +I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| +�� +Mh · · · M1 +�� +|+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� += +� +I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0| +�� +Mj · · · M1 +�� +|+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ +� +(N7) += p +� +I⊗(j−1) ⊗ ⟨oj| +�� +Mj−1 · · · M1 +�� +|+⟩ ⊗ I⊗(j−1)� +(N8) +< β +2 +� +I⊗(j−1) ⊗ ⟨oj| +�� +Mj−1 · · · M1 +�� +|+⟩ ⊗ I⊗(j−1)� +, +(N9) +where {ε, ζ} ∋ p < β/2 [63] depends on ij and oj, +• if ij and oj correspond to any permutation with no fixed point, +� +I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| +�� +Mh · · · M1 +�� +|+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� += +� +I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0| +�� +Mj · · · M1 +�� +|+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ +� +(N10) += p +� +I⊗(j−1) ⊗ ⟨oj| +�� +Mj−1 · · · M1 +�� +|+⟩ ⊗ I⊗(j−1)� +(N11) +< β +9 +� +I⊗(j−1) ⊗ ⟨oj| +�� +Mj−1 · · · M1 +�� +|+⟩ ⊗ I⊗(j−1)� +, +(N12) +where {µ, ν, ξ, o, π, τ, ϕ, χ, ψ} ∋ p < β/9 [63] depends on ij and oj. + +40 +Step 2. We now argue that the spectral radius of the matrix +� +I⊗(j−1) ⊗ ⟨oj| +�� +Mj−1 · · · M1 +�� +|+⟩ ⊗ I⊗(j−1)� +(N13) +is strictly bounded by 1 for 2 ≤ j ≤ h. It holds that the spectral radius of M is bounded by 1. While the first column +of |M| sums to 1, the sums of the other columns are strictly bounded by 1 [63]. As a result, the spectral radius of +Mj−1 · · · M1 is bounded by 1. The boundary condition |o⟩ does not affect this because it specifies a subset of columns +of the matrix Mj−1 · · · M1. In fact, it imposes a strict bound because this subset does not include the only column +summing to 1. Also the boundary condition ⟨+| does not affect the bound on the spectral radius. The boundary +condition means that Eq. (N13) is a sum of matrices. Each of these matrices comprises a disjoint subset of rows of +the matrix Mj−1 · · · M1. Because the spectral radius of Mj−1 · · · M1, the matrix comprising the whole set of rows, is +bounded by 1, so is the sum of the matrices comprising the disjoint subsets. +Step 3. 1 and β are the two leading eigenvalues of the first diagonal block N. They are non-degenerate. Because +the spectral radius of the matrix in Eq. (N13) is strictly bounded by 1, the eigenvalues of any other diagonal block +are strictly smaller than β. +The statement follows. +Appendix O: Proof of Result 4 +Result 4. The average of I2(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched +in Fig. 4 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ2D defined in Eq. (35). +Proof. We split the proof into four steps, following the structure of the proof of Result 1. The steps are overall very +similar to those of that proof. +Step 1. We rewrite EI2(A : B) in terms of expressions of the form of Eq. (93). As in one dimension, we make the +assumption that E log(X) = log(EX). Then, +EI2(A : B) = log +� +E tr +� +ϱ2 +AB +�� +− log +� +E tr +� +ϱ2 +A +�� +− log +� +Etr +� +ϱ2 +B +�� +. +(O1) +E tr +� +ϱ2 +A +� +, E tr +� +ϱ2 +B +� +, and E tr +� +ϱ2 +AB +� +can be written in the desired form [see Eqs. (62) and (63)]. +Step 2. We express EI2(A : B) in terms of the transfer tensors defined in Sec. V A and use Eq. (100) to map +contractions of two-dimensional tensor networks to multiplications of matrices. The latter is enabled by our definition +of subsystems A and B [see Fig. 4 (a)]. We have done this for E tr +� +ϱ2 +A +� +in graphical notation in Sec. V C [see Eqs. (104) +and (105)]. It is easy to confirm that +EI2(A : B) = log +� +⟨I2|T c +e T a +(12)T r +e T b +(12)|F2⟩ +� +− log +� +⟨I2|T c +e T a +(12)|F2⟩ +� +− log +� +⟨I2|T c+a+r +e +T b +(12)|F2⟩ +� +. +(O2) +Step 3. We expand EI2(A : B) in terms of the spectrum of Te with e ∈ S2, which we consider in App. M. Because +we know λ1 and λ2 as well as their algebraic and geometric multiplicities, we do not need Te to be diagonalizable. +Expanding T c +e and taking the limit c → ∞ yields +EI2(A : B) = log +� +⟨L1|T a +(12)T r +e T b +(12)|F2⟩ +� +− log +� +⟨L1|T a +(12)|F2⟩ +� +− log +� +⟨L1|T b +(12)|F2⟩ +� +, +(O3) +where we have used that ⟨I2|R1⟩ = 1. After expanding T r +e and using that |F2⟩ = |R1⟩, we have +I2(A : B) = log +� +⟨L1|T a +(12)|R1⟩⟨L1|T b +(12)|R1⟩ + λr +2⟨L1|T a +(12)|R2⟩⟨L2|T b +(12)|R1⟩ + O +� +rv−1λr +3 +�� +− log +� +⟨L1|T a +(12)|R1⟩ +� +− log +� +⟨L1|T b +(12)|R1⟩ +� +, +(O4) +where v denote the size of the largest Jordan block with respect to λ3. + +41 +Step 4. Finally, we can write EI2(A : B) in the form of Definition 1. With Λ = max +�� +λ2r +2 , rv−1λr +3 +�� +, +EI2(A : B) = log +� +1 + λr +2 +⟨L1|T a +(12)|R2⟩⟨L2|T b +(12)|R1⟩ +⟨L1|T a +(12)|R1⟩⟨L1|T b +(12)|R1⟩ + O +� +rv−1λr +3 +� +� +(O5) += λr +2 +⟨L1|T a +(12)|R2⟩⟨L2|T b +(12)|R1⟩ +⟨L1|T a +(12)|R1⟩⟨L1|T b +(12)|R1⟩ + O(Λ) +(O6) +≡ K exp +� +−r +ξ +� ++ O +� +exp +� +− r +χ +�� +, +(O7) +where +K = +⟨L1|T a +(12)|R2⟩⟨L2|T b +(12)|R1⟩ +⟨L1|T a +(12)|R1⟩⟨L1|T b +(12)|R1⟩ +(O8) +and +ξ = − +1 +log(λ2) = − +� +log +�dD3 − dD +d2D4 − 1 +��−1 += ξ2D > χ. +(O9) +This concludes the proof. +Appendix P: Proof of Result 5 +Result 5. The average of N(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched +in Fig. 4 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ2D defined in Eq. (35). +Proof. We split the proof into four steps, following the structure of the proof of Result 1. The steps are overall very +similar to those of the proof of Result 2 (see App. H). +Step 1. We rewrite EN(A : B) in terms of expressions of the form of Eq. (93). As in one dimension, with the +Hilbert-Schmidt inner product, +EN(A : B) = E tr +� +ϱ2 +AB +� ++ E tr +� +ϱ2 +A +� +tr +� +ϱ2 +B +� +− 2E tr[ϱAB(ϱA ⊗ ϱB)]. +(P1) +The right-hand side can be written in the desired form [see Eq. (H2)]. +Step 2. We express EN(A : B) in terms of the transfer tensors defined in Sec. V A and use Eq. (100) to map +contractions of two-dimensional tensor networks to multiplications of matrices. The latter is enabled by our definition +of subsystems A and B [see Fig. 4 (a)]. It is easy to confirm that +EN(A : B) = ⟨I4|T c +e T a +(12)T r +e T b +(12)|F4⟩ + ⟨I4|T c +e T a +(34)T r +e T b +(12)|F4⟩ − 2⟨I4|T c +e T a +(12)T r +e T b +(13)|F4⟩ +(P2) += ⟨I4|T c +e T a +(12)T r +e +� +T b +(12) + T b +(34) − 2T b +(13) +� +|F4⟩ +(P3) +≡ ⟨I4|T c +e AT r +e B|F4⟩, +(P4) +where we have defined +A = T a +(12) +and +B = T b +(12) + T b +(34) − 2T b +(13). +(P5) +Step 3. We expand EN(A : B) in terms of the spectrum of Te with e ∈ S4, which we consider in App. N. Because +we know λ1 and λ2 as well as their algebraic and geometric multiplicities, we do not need Te to be diagonalizable. +Expanding T c +e and taking the limit c → ∞ yields +EN(A : B) = ⟨L1|AT r +e B|F4⟩, +(P6) + +42 +where we have used that ⟨I4|R1⟩ = 1. After expanding T r +e and using that |F4⟩ = |R1⟩, we have +EN(A : B) = ⟨L1|A|R1⟩⟨L1|B|R1⟩ + λr +2 +w2 +� +µ=1 +⟨L1|A|R(µ) +2 ⟩⟨L(µ) +2 |B|R1⟩ + O(λr +3) +(P7) += λr +2 +w2 +� +µ=1 +⟨L1|A|R(µ) +2 ⟩⟨L(µ) +2 |B|R1⟩ + O(λr +3), +(P8) +where, in the second line, we have used that +⟨L1|B|R1⟩ = 0, +(P9) +which we prove in the following. +As in the proof of Result 2 (see App. H), we prove that ⟨L1|T b +t |R1⟩ does not depend on the two elements the +transposition t ∈ S4 acts upon. In fact, the proof follows from the proof of Eq. (H12) of that appendix. We just +need two additional considerations. First, the proof of Lemma 8 is not specific to the trivial permutation e ∈ S4. In +particular, Tρ exhibits an upper block triangular structure for any ρ ∈ S4. The first diagonal block is given by Tρ +with ρ ∈ S4, +� +⟨si|Tρ|sj⟩ +� +1≤i≤24 +1≤j≤24 += Tρ, +(P10) +which implies that +� +⟨si|T b +ρ |sj⟩ +� +1≤i≤24 +1≤j≤24 += T b +ρ. +(P11) +Second, the eigenvectors ⟨L1| and |R1⟩ of Te with e ∈ S4 arise from those of Te with e ∈ S4. That is, +|R⟩1 = s1 +and +� +⟨L1|si⟩ +� +1≤i≤7 = +� +1 +α +β − 1 +α +β − 1 +α +β − 1 +α +β − 1 +α +β − 1 +α +β − 1 +� +. +(P12) +⟨L1|T b +t |R1⟩ thus does not depend on the two elements the transposition t ∈ S4 acts upon because ⟨L1|T b +t |R1⟩ does +not. This implies that +⟨L1|B|R1⟩ = ⟨L1| +� +T b +(12) + T b +(34) − 2T b +(13) +� +|R1⟩ = 0. +(P13) +Step 4. Finally, we can write EN(A : B) in the form of Definition 1. That is, +EN(A : B) ≡ K exp +� +−r +ξ +� ++ O +� +exp +� +− r +χ +�� +, +(P14) +where +K = +w2 +� +µ=1 +⟨L1|A|R(µ) +2 ⟩⟨L(µ) +2 |B|R1⟩ +(P15) +and +ξ = − +1 +log(λ2) = − +� +log +�dD3 − dD +d2D4 − 1 +��−1 += ξ2D > χ. +(P16) +This concludes the proof. + diff --git a/1dE3T4oBgHgl3EQfnQrW/content/tmp_files/load_file.txt b/1dE3T4oBgHgl3EQfnQrW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ba10a8e6cc630de65b0515be816ed1a162556b4 --- /dev/null +++ b/1dE3T4oBgHgl3EQfnQrW/content/tmp_files/load_file.txt @@ -0,0 +1,2603 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf,len=2602 +page_content='Typical Correlation Length of Sequentially Generated Tensor Network States Daniel Haag,1, 2, 3, ∗ Flavio Baccari,1, 3 and Georgios Styliaris1, 3 1Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1, 85748 Garching, Germany 2Physik-Department, Technische Universität München, James-Franck-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1, 85748 Garching, Germany 3Munich Center for Quantum Science and Technology (MCQST), Schellingstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4, 80799 München, Germany (Dated: January 12, 2023) The complexity of quantum many-body systems is manifested in the vast diversity of their cor- relations, making it challenging to distinguish the generic from the atypical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This can be addressed by analyzing correlations through ensembles of random states, chosen so as to faithfully embody the relevant physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Here we focus on spins with local interactions, whose corre- lations are extremely well captured by tensor network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Adopting an operational perspective, we define ensembles of random tensor network states in one and two spatial dimensions that admit a sequential generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As such, they directly correspond to outputs of quantum circuits with a sequential architecture and random gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In one spatial dimension, the ensemble explores the entire family of matrix product states, while in two spatial dimensions, it corresponds to random isometric tensor network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We extract the scaling behavior of the average correlations between two sub- systems as a function of their distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using elementary concentration results, we then deduce the typical case for measures of correlation such as the von Neumann mutual information and a measure arising from the Hilbert–Schmidt norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We find for all considered cases that the typical behav- ior is an exponential decay (for both one and two spatial dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We observe the consistent emergence of a correlation length that only depends on the underlying spatial dimension and not the considered measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Remarkably, increasing the bond dimension leads to a higher correlation length in one spatial dimension but has the opposite effect in two spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' INTRODUCTION The behavior of correlations in quantum many-body systems is an inherently difficult problem to character- ize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Specifying a generic n-particle state requires ex- ponentially many parameters, a fact which reflects the enormous variety of correlations possible in the quantum realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Nonetheless, significant insights can be gained about the nature of correlations by utilizing random en- sembles of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A celebrated result along this direction shows that random states sampled uniformly from the full Hilbert space of an n-particle system typically ex- hibit strong correlations, as manifested by a volume law behavior for the entanglement entropy [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, there is by now clear evidence that the set of physically relevant states constitutes an exponentially small subset of the full Hilbert space of an n-particle system [6], bring- ing into question the relevance and utility of conclusions obtained under the assumption of uniform sampling from the full Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For quantum spin systems with local interactions, tensor network states have been exceedingly successful at capturing relevant properties [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' They exhibit an area law for the entanglement entropy by construction, and are, therefore, good candidates to represent many physically relevant many-body states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Their preeminent one-dimensional representatives, matrix product states (MPS), have been shown to represent faithfully ground states of gapped local Hamiltonians [8–10] and have given ∗ daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='haag@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='de rise to the complete classification of topological phases of matter in one dimension [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' MPS have been also generalized to their counterparts in two (or more) spa- tial dimensions, projected entangled pair states (PEPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While only a weaker link between local Hamiltonians and PEPS has been proven rigorously, two-dimensional PEPS are known to efficiently represent a wide class of strongly correlated states [7, 13], including states with power-law [14] and topological correlations [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The importance of defining ensembles of random tensor network states for the purpose of exploring typical prop- erties of physically relevant states has been recognized more than a decade ago [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' MPS ensembles have been utilized to gain insights into, among other things, the typ- icality of expectation values of local observables [18, 19], equilibration under Hamiltonian time evolution [20], the entropy of subsystems [21], non-stabilizerness [22], and, most relevant for this work, the behavior of correla- tions [23–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, correlation functions of ran- dom inhomogeneous MPS (that is, MPS whose local tensors can be different) were shown to exhibit almost surely an exponential decay [24, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A qualitatively similar behavior was observed also for correlation func- tions of translation-invariant MPS and PEPS with ran- dom Gaussian entries [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Instead of incorporating the randomness directly at the level of states, one can also consider random local Hamiltonians and examine their ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The typical behavior of correlations for this case was found to depend on the nature of random- ness, allowing for both long- and short-range correlated states [23, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Here, we approach the problem of typical correlations arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='04624v1 [quant-ph] 11 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Sequential generation of MPS and isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Each circle represents a site of the finalized state and boxes represent the isometries of the sequential generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (1a) Sequential generation of an MPS with physical dimension d and bond dimension D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The diamond indicates the origin of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (1b) Each isometry arises from a unitary matrix of U(dD), with input and output as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The (blue) ancillary system is initialized at the first step of the sequential generation, transferred along the process, and accumulated at the final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (2a) Sequential generation of an isoTNS with physical dimension d and bond dimension D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In addition to indicating the origin of the process, the diamond also indicates the orthogonality center of the isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (2b) Each isometry arises from a unitary matrix of U � dD2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Ancillary systems are initialized and eventually accumulated at the boundary of the isoTNS at different steps of the sequential generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' in random MPS and PEPS from a more operational point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We introduce families of inhomogeneous ran- dom tensor network states that arise from a sequential generation in a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Such ensembles are, by definition, directly connected to the study of quan- tum circuits with a sequential architecture and random gates, where each unitary gate is independently cho- sen randomly according to the uniform (Haar) measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the one-dimensional case, every MPS admits such a preparation [31], where the bond dimension dictates the number of overlapping qudits between any two succes- sive gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the two-dimensional setting, our ensemble can be understood as being uniform over the space of so-called isometric tensor network states (isoTNS) [32], which are PEPS with given bond dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In this case, the resulting family of random circuits is composed of two-dimensional circuits with local overlapping gates, each resembling a tile acting on a neighborhood of qu- dits [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Although isoTNS are a subfamily of PEPS, they are known to contain a rich variety of strongly cor- related states, such as topological models [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For the above ensembles, we study the scaling behavior of the average correlations between two subsystems as a function of their distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We then utilize this average behavior of correlations to deduce the typical case via concentration inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Instead of using well-known correlation functions, we perform the analysis using a measure of correlation arising from the Hilbert–Schmidt norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Although in a generic many-body setting such a measure might have undesirable properties, we show that it is particularly suited in the context of tensor network states because it bounds the trace distance as well as all connected correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For MPS, we also con- sider the Rényi-α mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given a technical conjecture, we compute the average correlations for all integer values of α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We then use those results to retrieve the von Neumann mutual information [35] via analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' First, we confirm analytically the common intuition that inhomogeneous MPS typically exhibit exponentially decaying correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show that a single common correlation length ξ1D persists among different measures of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We obtain a similar quantitative conclu- sion for two-dimensional isoTNS, where we observe the emergence of a different correlation length ξ2D that is also consistent among different measures of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Both lengths have a rather weak dependence on the underly- ing bond dimension D of the tensor network and remain short-range correlated for all values of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Surprisingly, however, ξ1D and ξ2D have exactly opposite behaviors when D is varied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ξ1D monotonically increases while ξ2D monotonically decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our findings also establish that exponentially decaying correlations are typical for the family of (inhomogeneous) isoTNS, and consequently for the random states produced by the corresponding quan- tum circuit architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II, we in- troduce our families of sequentially generated tensor net- work states and the main technical tools required to com- pute their average properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' III, we summarize our results for both MPS and isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV and V, we respectively discuss our findings for MPS and isoTNS in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lastly, we devote Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' VI to final observations and potential follow-ups to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' PRELIMINARIES In this section, we introduce the main technical con- cepts that will be needed throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II A, we review the relevant families of sequentially generated tensor network states in one and two dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II B is devoted to the measures of correlation we are interested to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II C, we explain how to compute averages with respect to the Haar mea- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lastly, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II D introduces the graphical notation we will use to present and prove our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Tensor Network States In one dimension, the preeminent tensor network struc- ture are matrix product states (MPS) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' An n-particle MPS with open boundary conditions and local (physical) dimension d is given by |ψ⟩ = � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',in ⟨L|A(1) i1 · · · A(n) in |R⟩|i1 · · · in⟩, (1) where A(j) ∈ CD×D, |L⟩ ∈ CD is the left boundary con- dition, and |R⟩ ∈ CD is the right boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' D is called the bond dimension of the MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A commonly used graphical notation for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (1) is |ψ⟩ = , (2) where vertical (red) legs represent physical space indices � Cd� , and (blue) legs represent bond space indices � CD� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From its definition, it might not be evident how an MPS can be prepared because each tensor A does not necessarily correspond to a physical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, the representation of an MPS in terms of tensors is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This can be resolved by imposing a convenient canonical form [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Any MPS in such a canonical form can be seen as a state generated sequentially by applying unitary matrices U (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , U (n) ∈ U(dD) to a product state initialized in |0⟩⊗n for the physical space and |0⟩ for the bond space [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The resulting state is given by |ψ⟩ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (3) Note that the final site has dimension dD, while all other sites have dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we will see later, the final site will not play a significant role in our analysis, making its different dimension not an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (1a), we sketch an equivalent representation of sequential generation, in terms of isometries instead of unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The family of MPS is thus equivalent to states re- sulting from quantum circuits that have a sequential architecture and act on input product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The ar- chitecture is a consequence of the connectivity of the MPS network [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (1a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In this picture, larger bond dimensions translate to wider gates, each acting on 1+⌈logd(D)⌉qudits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For example, for D = d2 and n = 4, one has |ψ⟩ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (4) Naturally, using this correspondence, all of our results can be translated to the language of quantum circuits with the described architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Projected entangled-pair states (PEPS) are the gener- alization of MPS to two (or more) dimensions [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Be- cause no simple generalization of the sequential gener- ation of MPS to arbitrary PEPS is known, we restrict ourselves to the rich family of two-dimensional isometric tensor network states (isoTNS), which were first defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [32] (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Much like MPS, isoTNS can be generated sequentially by applying unitary matrices to a product state initial- ized in |0⟩⊗mn for the physical space [33], where m de- notes the number of rows and n the number of columns of the underlying rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will use the sequential generation sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2a), which is a generalization of the one proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Each box corresponds to an isometry that arises from a unitary matrix U (i,j) ∈ U � dD2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, isometries in the bulk can be drawn as , (5) as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The diamond in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2a) in- dicates the so-called orthogonality center of the isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Its row and column constitute the orthogonality hyper- surface, which can be treated like an MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, if an operator is supported only on the orthogonality hyper- surface, its expectation value with respect to the isoTNS reduces to that of the underlying MPS [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Although isoTNS of a given bond dimension form by definition only a subclass of PEPS, they are known to contain states with a rich structure of correlations, such as topological mod- els [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' On top, their properties make isoTNS a suitable candidate for studying correlations analytically, which is otherwise a generally challenging task in more than one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IsoTNS correspond to quantum circuits on a two- dimensional grid with local overlapping gates, which now resemble tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Increasing bond dimension translates to larger tile sizes and overlaps, as in the MPS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The corresponding architecture is dictated by the connectiv- 4 ity of the isoTNS network [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2a)], and it is te- dious (although straightforward), which is why we refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [33] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Quantifying Correlations Correlations express that knowledge about one sub- system can convey information about another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' They are quantified by different measures that frequently arise from an information-theoretic perspective and are based on operationally motivated tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A prime example is the von Neumann mutual information [35] I(A : B) = S(ϱA) + S(ϱB) − S(ϱAB), (6) where S(ϱ) = − tr[ϱ log(ϱ)] (7) is the von Neumann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A and B are two disjoint subsystems of a larger system, and ϱA and ϱB denote the marginals of ϱAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The von Neumann mutual information captures the total (classical and quantum) amount of cor- relations between A and B, as it is equal to the minimum rate of randomness required to asymptotically turn ϱAB into a product state [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is also a non-negative quan- tity and non-increasing under local operations [35], both desirable properties for a measure of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The latter means that a quantum channel [35] acting on A or B alone (for example, by discarding part of a subsystem) cannot increase I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Unfortunately, the analytical treatment of the von Neumann mutual information is im- practical because computing the logarithm of ϱ generally requires the knowledge of its full spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To overcome this issue, an alternative is to consider a particular Rényi-α generalization of the mutual informa- tion Iα(A : B) = Sα(ϱA) + Sα(ϱB) − Sα(ϱAB), (8) where Sα(ϱ) = 1 1 − α log[tr(ϱα)] (9) is the Rényi-α entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As is apparent from the defini- tion, for integer values of α, its evaluation is considerably simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The Rényi-α mutual information has been inves- tigated in the context of conformal field theories [39–41], free fermions [42], and quantum dynamics [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will use later that the α → 1 limit of Iα(A : B) recovers the von Neumann mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The mentioned positive aspects notwithstanding, unlike the von Neu- mann mutual information, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (8) does not arise from a (generalized) divergence [45], and the Rényi-α mutual information can be negative [46, 47] and increasing under local operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is thus hard to interpret it as a proper measure of correlation in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Nevertheless, for cer- tain families of initial states (see, for example, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [44]) monotonicity and non-negativity can be restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Hence- forth, we will mostly focus on the case of α = 2, but we will also consider an analytic continuation on positive in- teger values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we will show, in the present context of tensor network states, the case of α = 2 appropriately captures the decay of correlations at large distances be- tween subsystems A and B with little effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In addition to the previous quantities, we would also like to probe the trace distance T(A : B) = 1 2∥ϱAB − ϱA ⊗ ϱB∥1, (10) where ∥ · ∥p denotes the Schatten p-norm [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For an operator X, the Schatten p-norm is given by ∥X∥p = tr �� X†X �p/2�1/p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (11) T(A : B) has a well-known operational interpretation, as it quantifies the optimal distinguishability between ϱAB and the product of its marginals ϱA⊗ϱB by a two-element generalized (global) measurement [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Moreover, the trace distance upper bounds the (properly normalized) connected correlation function [45]: T(A : B) ≥ 2|⟨MA ⊗ MB⟩ − ⟨MA⟩⟨MB⟩| ∥MA∥∞∥MB∥∞ (12) Although the bound can be tight, the two quantities are different whenever product measurements are ineffective in distinguishing ρAB from ρA ⊗ ρB, a fact used in quan- tum data hiding [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As one expects from its operational interpretation, the trace distance satisfies the monotonicity property under local operations [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, T(A : B) is usually hard to compute exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will now argue that investigating N(A : B) = ∥ϱAB − ϱA ⊗ ϱB∥2 2 (13) meaningfully probes T(A : B) for tensor network states with constant bond dimension, all while being much sim- pler to treat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In general, for mixed many-body states, the two mea- sures can have vast disagreement because it holds [48] that ∥X∥2 ≤ ∥X∥1 ≤ � rank(X)∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (14) Both bounds are tight, and the upper bound is saturated for X ∝ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As such, for arbitrary mixed states of an exponentially large Hilbert space, the factor rank(X) can render the upper bound useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Crucially, in this work we investigate (random) tensor network states with fixed bond dimension D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let ∂R denote the boundary of a system R and |∂R| its size (number of sites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The ranks of ϱA and ϱB are respectively upper bounded by D|∂A| and D|∂B|, and that of ϱAB by D|∂A|+|∂B| [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, rank(ϱAB − ϱA ⊗ ϱB) ≤ 2D|∂A|+|∂B|, (15) 5 yielding the bound 1 2 � N(A : B) ≤ T(A : B) ≤ � D|∂A|+|∂B| 2 � N(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (16) For MPS (one dimension) with connected subsystems A and B, the bound reads 1 2 � N(A : B) ≤ T(A : B) ≤ D2 √ 2 � N(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (17) That is, the bound is independent of the sizes of A and B, unlike in the generic case of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This suggests that, for reasonably small bond dimension, N(A : B) is a reliable probe of correlations [as quantified by T(A : B)] between subsystems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will expand on this point later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' k-Fold Twirl Let X be an operator acting on (Cq)⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The k-fold twirl of X with respect to the Haar measure on the uni- tary group U(q) is defined [51–53] as T (k) U (X) = � dU U ⊗kX � U †�⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (18) One can employ the Schur–Weyl duality for unitary groups to show [51, 54] that T (k) U (X) = � σ,τ∈Sk Wg � στ −1, q � P (q) σ tr � X � P (q) τ �T � , (19) where P (q) π : v1 ⊗ · · · ⊗ vk �→ vσ−1(1) ⊗ · · · ⊗ vσ−1(k) (20) is the representation of π ∈ Sk on (Cq)⊗k, where Sk is the symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Wg � στ −1, q � = � G−1� στ [55] is the Weingarten function, where G ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' denotes the Gram matrix whose entries are given by Gστ = tr � P (q) σ � P (q) τ �T � = q#(στ −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (21) Above, #(π) counts the number of cycles in the decom- position of π ∈ Sk into disjoint cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, Wg(π, q) depends only on the conjugacy class of π [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A, we show how to obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (19) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (18) by using a result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Graphical Notation In this section, we introduce the graphical notation used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To keep the images compact, we employ the operator-vector correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let {|i⟩} denote the standard basis of Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, the operator- vector correspondence [48] is defined by vec(|i⟩⟨j|) = |i⟩ ⊗ |j⟩ (22) and extended linearly to the vector space at large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we consider the standard (product) basis to be fixed, we do not distinguish between tensors (as mul- tidimensional arrays) and their basis-independent coun- terparts (such as vectors and operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let X be an operator acting on (Cq)⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the operator-vector correspondence, we denote it by = vec(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (23) Note that the orientation of the legs does not have any meaning in our images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, = = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (24) When we need the transpose of an operator, we will ex- plicitly use = vec � XT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (25) As such, when we contract two operators X and Y , we mean the trace of their product: = tr(XY ) (26) Let us state the two most prominent operators we will come across.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will see = vec � P (q) σ � , (27) where the horizontal (green) leg is permutation valued, and = vec � |0⟩⟨0|⊗k� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (28) Their relevant contractions are = tr � |0⟩⟨0|⊗kP (q) σ � = 1 (29) and = tr � P (q) σ P (q) τ � = q#(στ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (30) Moving forward, we will not explicitly write the oper- ator vec as it shall be clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We investigate average correlations between two sub- systems A and B as a function of their (horizontal) distance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A and B respectively stretch across a and b consecutive (horizontal) sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (a) The diamond indicates the origin of the sequential generation of the MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (b) In addition to indicat- ing the origin of the sequential generation, the diamond also indicates the orthogonality center of the isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For now, we restrict ourselves to A and B that touch the orthogonality hypersurface and stretch across h consecutive vertical sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With the definition of the Weingarten matrix, = Wg � στ −1, q � , (31) we can then write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (19) as T (k) U (X) = , (32) where the contraction of two green legs corresponds to a summation over the permutations of Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' SUMMARY OF RESULTS In this paper, we analyze the average behavior of cor- relations in random tensor network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Through the average, we obtain conclusions about the generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our work focuses on the disordered case, that is, the case where each local tensor is independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our setting can be equivalently understood as an investigation of corre- lations in states resulting from quantum circuits with a sequential architecture and random gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In one dimension, generic MPS are known to exhibit exponentially decaying correlations in the translation- invariant case [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This is due to the fact that injectivity is a generic property [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' On the other hand, injectivity alone is not enough to guarantee exponential decay of cor- relation for an inhomogeneous sequence of tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Nev- ertheless, the exponential decay of correlations is widely expected to persist without translational invariance but has never been rigorously studied so far in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In two (or more) dimensions, the landscape of correla- tions is much richer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For instance, already in two dimen- sions, certain PEPS corresponding to thermal states of classical models are known to exhibit power-law correla- tions [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Moreover, prominent topological states, such as quantum double models [57] (which include the toric code) and string-net models [16, 17], admit a description in terms of PEPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' On the other hand, for translation- invariant PEPS whose tensors’ entries are drawn from a Gaussian measure, it is known that correlations typically decay exponentially [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Computing correlations in higher-dimensional systems usually poses a significant challenge because they can be mediated through different paths connecting the two sub- systems of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Here, we restrict our analysis to two-dimensional isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This rich class of tensor net- work states is relevant in both the analytical and the numerical context [58–62], all while admitting a simple physical interpretation through sequential generation [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Moreover, its mathematical properties make the analytical study of correlations in two dimensions tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For isoTNS, it is expected that correlations between two subsystems decay exponentially if they are both on the orthogonality hypersurface because the calculation reduces to the contraction of an MPS [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Nonetheless, isoTNS can represent a rich variety of topological mod- els, as all string-net models admit an exact and explicit description in terms of isoTNS [34] (on the appropriate underlying lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This motivates us to study the typ- ical behavior of correlations in isoTNS, particularly be- tween subsystems that extend beyond the orthogonality hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To investigate the decay of correlations in our two fam- ilies of random tensor network states, one must specify the ensembles to draw from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Here we adopt an oper- ational perspective and relate our measures of random- ness directly to the sequential generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Be- cause that is defined with respect to isometries, one can incorporate randomness at the level of the under- lying unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A natural choice is to draw each unitary from the Haar measure on the appropriate uni- tary group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This approach was introduced for MPS in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [18] (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [19]), and it can directly be applied to higher-dimensional tensor network states that admit a sequential generation, such as isoTNS, yielding normal- ized states by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Although one can sample random translation-invariant states with this method, we investigate the disordered case by drawing each unitary matrix independently from the Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we are interested in the decay of correlations, we focus on computing average correlations between two subsystems A and B as a function of their distance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For 7 random MPS and isoTNS, we consider subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In both cases, A and B stretch across a and b consecu- tive (horizontal) sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (b), A and B touch the orthogonality hypersurface and stretch across h vertical sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will relax this condition later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For all of the measures of correlation we study, we find that the average with respect to the considered ensemble of states decays exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We formalize this type of behavior in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let C(A : B) denote a measure of correla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We say that the average of C(A : B) with respect to a given ensemble of random states decays exponentially if EC(A : B) = K exp � −r ξ � + O � exp � − r χ �� , (33) where K is constant with respect to r, and ξ > χ is the average correlation length for C(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Remarkably, we find that a single average correlation length persists throughout the different families of mea- sures of correlation and that it depends only on the un- derlying spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We later pinpoint the origin of this behavior to the invariance of a spectral gap of the corresponding family of transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For MPS, ξ = − � log � dD2 − d d2D2 − 1 ��−1 ≡ ξ1D, (34) and for isoTNS, ξ = − � log �dD3 − dD d2D4 − 1 ��−1 ≡ ξ2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35) Note that the average correlation length for MPS co- incides with that for isoTNS for d → dD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we will see later, this seemingly small modification changes the qualitative behavior substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Before moving to the detailed presentation of our methods and results, we briefly comment on the consid- ered measures of correlation and the implications of our findings, first for MPS and then for isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Results in One Dimension (MPS) In one dimension, we compute the averages of the Rényi-2 mutual information I2(A : B), the 2-norm ex- pression N(A : B), and the von Neumann mutual infor- mation I(A : B) (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II B for the definitions of the measures of correlation), with subsystems A and B as sketched in Fig 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We find that the averages decay exponentially as spec- ified in Definition 1 with the same correlation length ξ1D (see Results 1, 2, and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The derivation for I(A : B) relies on a technical conjecture (see Conjecture 1), which we will discuss in detail later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In addition, we show that the same conjecture is enough to assert that ξ1D is also the average correlation length for Iα(A : B) for any inte- ger value of α ≥ 1 (see Corollary 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In short, the same average correlation length ξ1D per- sists across different measures of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Interest- ingly, ξ1D depends very weakly on the bond dimension because, for d, D ≥ 2, ξ1D = � log � d ζ1D(d, D) ��−1 (36) with 3 4 ≤ ζ1D(d, D) < 1 (37) is monotonically increasing with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, it holds that limD→∞ ξ1D = 1/ log(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we are concerned with random tensor network states, ξ1D is obtained after averaging over realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is then natural to ask if exponentially decaying corre- lations are typical and, if so, what is the typical correla- tion length for an individual realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This motivates the investigation of the concentration of the distribution around its average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, we will show that it is exponentially unlikely in r that N(A : B) and I(A : B) decay slower than with ξ1D (see Corollaries 1 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our result for N(A : B) allows us to deduce that the average of the trace distance T(A : B) decays at least exponen- tially with correlation length ξ ≤ 2ξ1D, and it leads to a concentration result for T(A : B) (see Corollary 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Results in Two Dimensions (isoTNS) In two dimensions, we compute the averages of the Rényi-2 mutual information I2(A : B) and the 2-norm expression N(A : B), where subsystems A and B as sketched in Fig 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, we find that the averages decay exponentially as specified in Definition 1 with the same average correlation length ξ2D (see Results 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The correlation length ξ2D displays a qualitatively dif- ferent dependence on the bond dimension that is albeit also rather weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, for d, D ≥ 2, ξ2D = � log � d ζ2D(d, D) ��−1 (38) with 0 < ζ2D(d, D) ≤ 8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (39) In contrast to its one-dimensional counterpart, ξ2D is a monotonically decreasing function of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As such, perhaps somewhat surprisingly, the largest correlation length is achieved for D = 2, which is still rapidly de- caying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For N(A : B), we can extend the applicability of our results to any size and shape of subsystems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 8 The decay is at least exponential with correlation length ξ = ξ2D (see Corollary 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We furthermore prove a con- centration result for N(A : B) expressing that it is highly unlikely that N(A : B) decays slower than with ξ2D (see Corollary 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This also allows us to draw a similar con- clusions about the behavior of T(A : B) (see Corollary 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' CORRELATIONS IN ONE DIMENSION In this section, we state and discuss the results for random MPS summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' III A in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Before doing that, we develop the tools behind our proofs in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A and IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV C, we compute the average of I2(A : B), and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV D, we investigate the decays of N(A : B) and T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we discuss the behavior of I(A : B) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' When computing the averages of measures of corre- lation for random MPS, we will exploit a simplification with respect to the scenario depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In- stead of allowing for an arbitrary number of sites be- fore subsystem A, we prove our statements in the limit c → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' J, this does not constitute a limitation because the c initial sites do not affect the decay of correlations and, therefore, neither the average correlation length ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Furthermore, we will see that the f sites after subsystem B do not play a role in the com- putation of average correlations, as it is expected for any sequentially generated state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Transfer Matrices The key challenge for computing the average of each measure of correlation will be evaluating multiple expres- sions of the form tr � PE|ψ⟩⟨ψ|⊗k� , (40) where P = � P (d) e �⊗c ⊗ � P (d) α �⊗a ⊗ � P (d) e �⊗r ⊗ � P (d) β �⊗b ⊗ � P (d) e �⊗(f−1) ⊗ P (dD) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (41) The permutation α ∈ Sk acts on the sites comprising subsystem A, while β ∈ Sk acts on the sites comprising B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The exact forms of α and β as well as the number of required replicas k depends on the considered measure of correlation and will be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It shall also be- come clear why sites belonging to neither A nor B are acted upon by the trivial permutation e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the fol- lowing, we show that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40) for random MPS reduces to multiplying matrices Tρ ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' with ρ ∈ Sk whose definition will be all but natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because their role is analogous to the known transfer matrices mediating cor- relations, we will also adopt this term here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Before introducing the transfer matrices, we must an- alyze E|ψ⟩⟨ψ|⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, let us define V (j) = U (j) ⊗ U (j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (3), |ψ⟩⟨ψ| = (42) and |ψ⟩⟨ψ|⊗k = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (43) By computing the k-fold twirl [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (32)], we obtain the building block = � dU (44) = , (45) where the (green) dot represents a Kronecker delta on three permutation indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Note that we have not drawn the contraction of a permutation matrix with |0⟩⟨0|⊗k because it is trivial by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of a random MPS is then given by E|ψ⟩⟨ψ|⊗k = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (46) We could, in principle, work with the building block above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, it is not convenient to have dangling bond (blue) legs whose dimension grows with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By cutting permutation-valued (green) legs instead, we ob- tain a building block with fixed dimension for fixed k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With that building block, the average of a random MPS is given by E|ψ⟩⟨ψ|⊗k = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (47) The entries of the initial vector ⟨Ik| ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' are given by = = 1, (48) 9 where we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The tensors in the bulk are given via = , (49) and the final tensor is given via = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (50) Computing an expression of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40) cor- responds to contracting each S with some P (d) ρ , which leads us to the promised definition of a transfer matrix Tρ ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='. Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (30), its entries are given by = = (51) = � σ∈Sk Wg � στ −1, dD � d#(σρ)D#(σθ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (52) We define Tρ with respect to the basis defined by the map si �→ ei, where si is the ith element of Sk = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , sk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' }, and {ei} is the standard basis of Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='. In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' D, we find that Tρ is block triangular if the elements of Sk are ordered in a certain way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As alluded to in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (41), the final tensor S′ will be contracted with the trivial permutation e ∈ Sk in our computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The final vector |Fk⟩ ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' is thus defined via = = (53) = � σ∈Sk Wg � στ −1, dD � (dD)#(σe) = δeτ , (54) where we have used the definition of the Weingarten func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the definitions of Te and |Fk⟩, it is easy to con- firm that Te|Fk⟩ = |Fk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Graphically, this implies the simplification = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (55) From this, it also follows that E|ψ⟩⟨ψ|⊗k is properly nor- malized: tr � E|ψ⟩⟨ψ|⊗k� = ⟨Ik|T n−1 e |Fk⟩ = ⟨Ik|Fk⟩ = 1 (56) With what we have laid out above, we can write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40) in terms of transfer matrices: tr � PE|ψ⟩⟨ψ|⊗k� = ⟨Ik|T c e T a αT r e T b β|Fk⟩ (57) We provide a simple Mathematica package [63] that defines Tρ with ρ ∈ Sk for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , 20} according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The package relies on the package provided by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [64] for evaluating the Weingarten function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Estimating the Decay of Correlations The decay of the average of each measure of correla- tion is necessarily determined by the r sites separating subsystems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we will see in the following sec- tions, this will, for each measure, translate to a simple statement in terms of the just-defined transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we will find that the decay of correlations is determined by T r e with e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Taking this as a fact for now, we connect the decay of correlations with the spectrum of Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The spectrum of Te depends on k because k determines its dimension and entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Still, we can make general statements about Te for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we will prove the following statements in Apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The eigenvalues of Te with e ∈ Sk are non-negative for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Te with e ∈ Sk is diagonalizable for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the dis- tinct eigenvalues of Te with e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, λ1 = 1 and it is non-degenerate for any k ≥ 2 if d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given the statements above, we can expand T r e as T r e = |R1⟩⟨L1| + λr 2 w2 � µ=1 |R(µ) 2 ⟩⟨L(µ) 2 | + O(λr 3), (58) where |R(µ) i ⟩ denotes the µth right eigenvector corre- sponding to λi, ⟨L(µ) i | denotes the µth left eigenvector corresponding to λi, and wi denotes the degeneracy of λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The asymptotic decay of correlations is thus deter- mined by the subleading eigenvalue λ2 of Te, and the average correlation length is given by ξ = − 1 log(|λ2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (59) The argument behind this is similar to one known from the analysis of correlations in translation-invariant MPS [56, 65], where the decay is determined by the sub- leading eigenvalue of the relevant transfer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Rényi-2 Mutual Information We start our analysis of correlations in random MPS with the simplest case, namely the computation of the average of the Rényi-2 mutual information I2(A : B) = log � tr � ϱ2 AB �� − log � tr � ϱ2 A �� − log � tr � ϱ2 B �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (60) The analytical treatment turns out to be comparatively simple if one assumes that E log(X) = log(EX), as is frequently done in this context [66–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will not make this assumption further below when we study the von Neumann mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The analysis there will require transfer matrices Tρ with ρ ∈ Sk for all k ≥ 2, while ρ ∈ S2 will suffice here because only the averages of purities are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our first result is summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of I2(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We split the proof into four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The exact same structure will also appear in the proofs for the other mea- sures of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, this proof serves as the sim- plest example and a point of reference for later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We rewrite EI2(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, we make the as- sumption that E log(X) = log(EX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, EI2(A : B) = log � E tr � ϱ2 AB �� − log � E tr � ϱ2 A �� − log � E tr � ϱ2 B �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (61) E tr � ϱ2 A � , E tr � ϱ2 B � , and E tr � ϱ2 AB � can already be written in the desired form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For example, E tr � ϱ2 AB � = tr � PE|ψ⟩⟨ψ|⊗2� (62) with P = � P (d) e �⊗c ⊗ � P (d) (12) �⊗a ⊗ � P (d) e �⊗r ⊗ � P (d) (12) �⊗b ⊗ � P (d) e �⊗(f−1) ⊗ P (dD) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (63) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We express EI2(A : B) in terms of the transfer matrices defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given the previous step, it is easy to confirm that EI2(A : B) = log � ⟨I2|T c e T a (12)T r e T b (12)|F2⟩ � − log � ⟨I2|T c e T a (12)|F2⟩ � − log � ⟨I2|T c+a+r e T b (12)|F2⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (64) Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EI2(A : B) in terms of the spectrum of Te with e ∈ S2 [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (58)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the relevant expressions for the Weingarten function, it is evident [69] that Te = � � � � 1 d2D − D d2D2 − 1 0 dD2 − d d2D2 − 1 � � � � (65) is diagonalizable with λ1 = 1 and λ2 = dD2 − d d2D2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (66) Expanding T c e and taking the limit c → ∞ yields EI2(A : B) = log � ⟨L1|T a (12)T r e T b (12)|F2⟩ � − log � ⟨L1|T a (12)|F2⟩ � − log � ⟨L1|T b (12)|F2⟩ � , (67) where we have used that ⟨I2|R1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' After expanding also T r e and using that |F2⟩ = |R1⟩, we have EI2(A : B) = log � ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ + λr 2⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ � − log � ⟨L1|T a (12)|R1⟩ � − log � ⟨L1|T b (12)|R1⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (68) Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we can write EI2(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, EI2(A : B) = log � 1 + λr 2 ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ � (69) = λr 2 ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ + O � λ2r 2 � (70) ≡ K exp � −r ξ � + O � exp � −2r ξ �� , (71) where K = ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ (72) and ξ = − 1 log(λ2) = − � log � dD2 − d d2D2 − 1 ��−1 = ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (73) This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 11 As we discussed earlier, the Rényi-2 mutual informa- tion is lacking many of the desirable properties that a sound measure of correlation ought to fulfill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' On top, our computation simplifies considerably because we are using the assumption that E log(X) = log(EX), which amounts to ignoring statistical fluctuations in the dif- ferent realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the following section, we will see that N(A : B) decays exponentially with the same av- erage correlation length ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will furthermore show that N(A : B) concentrates around its average, provid- ing evidence that fluctuations can be safely ignored in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Trace Distance and 2-Norm In this section, we investigate average correlations as quantified by the trace distance T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As anticipated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II B, this is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, as laid out there, the 2-norm expression N(A : B) reliably esti- mates T(A : B) for the case of random MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Hence, we compute the average of N(A : B) = ∥ϱAB − ϱA ⊗ ϱB∥2 2 (74) = tr � ϱ2 AB � + tr � ϱ2 A � tr � ϱ2 B � − 2 tr[ϱAB(ϱA ⊗ ϱB)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (75) Because of its connection to the Hilbert–Schmidt in- ner product, the average of N(A : B) can be computed without any simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Making use of the transfer-matrix techniques introduced above, we prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Result 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of N(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Sketch of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The proof follows the same procedure as that of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Here, we sketch the main steps and refer to App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' H for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Step 1, we write EN(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The second summand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (75) requires permutations of S4 because E tr � ϱ2 A � tr � ϱ2 B � = tr � PE|ψ⟩⟨ψ|⊗4� (76) with P = � P (d) e �⊗c ⊗ � P (d) (12) �⊗a ⊗ � P (d) e �⊗r ⊗ � P (d) (34) �⊗b ⊗ � P (d) e �⊗(f−1) ⊗ P (dD) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (77) The first summand and the third summand respectively require only permutations of S2 and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Step 2, we thus write EN(A : B) in terms of transfer matrices Tρ with ρ ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This means that the average correlation length is de- termined by the subleading eigenvalue of Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Step 3, we find that λ1 = 1 and λ2 = dD2 − d d2D2 − 1, (78) just like for Te with e ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The former is non- degenerate, while the degeneracy of the latter is given by the number of transposition in S4, w2 = �4 2 � = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (79) Thus, the average correlation length for N(A : B) co- incides with that for I2(A : B), as we conclude in Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The above result establishes the exponential decay of the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, one is usually interested in know- ing if typical instances are expected to have the same exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This can be easily established by Markov’s inequality because N(A : B) is non-negative and its average decays to zero as a function of the dis- tance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a), sufficiently large r, and all 0 < ε < 1, the random MPS ensemble satisfies Pr � N(A : B) ≥ K exp � −(1 − ε)r ξ1D �� ≤ exp � − εr ξ1D � , (80) where K is constant with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By Result 2, for sufficiently large r, we can bound EN(A : B) ≤ K exp(−r/ξ1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because N(A : B) is non-negative, by Markov’s inequality, we have, for η > 0, Pr � N(A : B) ≥ ηK exp � − r ξ1D �� ≤ Pr[N(A : B) ≥ ηEN(A : B)] (81) ≤ 1 η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (82) The result follows with η = exp(εr/ξ1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The above corollary reflects that it is exponentially unlikely in r that N(A : B) decays slower than with the average correlation length ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we have already established that the average case exhibits an exponential decay with correlation length ξ1D, the average case is also typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By combining the above result with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (17), we can now also bound the correlation length for T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='9 ξ 6 8 10 12 14 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='7 d = 2 d = 3 d = 4 d = 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Numerically obtained average correlation length ξ for different d and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The data points are obtained by fitting the average value of I(A : B) against r ∈ {5, 7, 9, 11, 13, 15} for a = b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The sample size of 10 000 suffices for the error bars to lie within the plot points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The opaque curves correspond to ξ1D [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34)] Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a), sufficiently large r, and all 0 < ε < 1, the random MPS ensemble satisfies Pr � T(A : B) ≥ K exp � −(1 − ε)r 2ξ1D �� ≤ exp � − εr ξ1D � , (83) where K is constant with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that ET(A : B) ≤ � E[T(A : B)]2 (84) ≤ D2 2 � EN(A : B) (85) ≤ K exp � − r 2ξ1D � , (86) where, in the last line, we have assumed r to be suffi- ciently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The result follows as in the proof of Corol- lary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, with overwhelming probability, correlations as quantified by T(A : B) decay exponentially with ξ ≤ 2ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Von Neumann Mutual Information The fact that I2(A : B) and N(A : B) have the same average correlation length ξ1D motivates the question whether other measures of correlation behave similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In this section, we will provide compelling evidence that ξ1D is indeed the average correlation length also for the von Neumann mutual information I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We start by numerically investigating the behavior of I(A : B) for random MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We have generated MPS ac- cording to our measure, computed the average of I(A : B), and extracted the average correlation length from fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 3 shows, the numerically obtained average correlation length coincides well with ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It should be noted that we have set c = 0 for our numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We discuss in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' J why this does not affect the aver- age correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For more details on our numerical analysis, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We now turn to the analytical computation of the av- erage of I(A : B) = tr[ρAB log(ρAB)] − tr[ρA log(ρA)] − tr[ρB log(ρB)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (87) To be able to make use of the transfer-matrix tech- niques introduced above, we employ two replica tricks to write EI(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' First, we write S(ρ) as the α → 1 limit of Sα(ρ): S(ρ) = lim α→1 1 1 − α log[tr(ϱα)] (88) Second, instead of assuming again that E log(X) = log(EX), we use E log(X) = lim v→0 1 v log(EXv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (89) We are thus dealing with expressions of the form tr(PE|ψ⟩⟨ψ|⊗vα), which require transfer matrices Tρ with ρ ∈ Svα (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This means that knowing the spec- trum of Te with e ∈ Svα for all vα ≥ 2 allows us to draw conclusions about the decay of the average of I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While this is in principle a daunting task, we are able to prove several properties of the transfer matrix Te with e ∈ Sk for any k ≥ 2 (see Propositions 1, 2, and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' D, we furthermore show that Te with e ∈ Sk has eigenvalue µ2 = dD2 − d d2D2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (90) for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Its degeneracy is at least v2 = �k 2 � , (91) the number of transpositions in Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We conjecture that µ2 is the subleading eigenvalue of Te with e ∈ Sk for any k ≥ 2 and that it has degeneracy v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We know this conjecture to hold for k ∈ {2, 3, 4}, and we have numerical evidence suggesting so for k ∈ {5, 6, 7} [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We were not able to prove the statement outright, but in the following we argue that it is the only missing step to show that ξ1D is the average correlation length for I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let us state the conjecture formally below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 13 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the dis- tinct eigenvalues of Te with e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, λ2 = µ2 with degeneracy w2 = v2 for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, the properties of Te with e ∈ Sva that are relevant for determining the decay of correlations are independent of vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' I, we argue that the replica limit does not affect this and prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, the average of I(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We provide a concentration result also for I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement and its proof are identical to that for N(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is exponentially unlikely in r that I(A : B) decays slower than with the average correlation length ξ1D, and the average case is also typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, for subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a), sufficiently large r, and all 0 < ε < 1, the random MPS ensemble satisfies Pr � I(A : B) ≥ K exp � −(1 − ε)r ξ �� ≤ exp � −εr ξ � , (92) where K is constant with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Another corollary of Result 3 is that ξ1D is also the average correlation length for Iα(A : B) for any integer value of α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We prove also this statement in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, for any integer value of α ≥ 1, the average of Iα(A : B) with respect to the ran- dom MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Defini- tion 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, let us summarize the reason behind the per- sistent appearance of the average correlation length ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In all of the examined cases, the asymptotic behavior of the correlations was determined by the asymptotic de- cay of T r e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Although the transfer matrix Te with e ∈ Sk does depend on the number of replicas k, its asymptotic decay does not (given Conjecture 1), resulting in a com- mon average correlation length ξ1D across measures of correlation with different complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' CORRELATIONS IN TWO DIMENSIONS In this section, we state and discuss in more detail the results for random isoTNS summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V A, we develop the two-dimensional analog to the transfer matrices introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A, the tool behind our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V C, we compute the average FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We investigate average correlations in random isoTNS between two subsystems A and B as a function of their hori- zontal distance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A and B respectively stretch across a and b consecutive horizontal sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In addition to indicating the origin of the sequential generation, the diamond also indicates the orthogonality center of the isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (a) For Results 4 and 5, we consider A and B that touch the orthogonality hyper- surface and stretch across h consecutive vertical sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (b) We will provide an additional result for the 2-norm expression N(A : B) for arbitrary (but fixed) A and B that do not need to touch the orthogonality hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' of I2(A : B), and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V D, we investigate the decays of N(A : B) and T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, we prove our statements in the limit c → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show the exponential decay of the average for each considered measure of correlation and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) as a function of the distance r between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we prove that I2(A : B) and N(A : B) have a common average correlation length that is independent of the sizes of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Moreover, thanks to the more amenable properties of N(A : B), we are able to show that average correlations decay expo- nentially also for subsystems A and B that do not have to touch the orthogonality hypersurface [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Transfer Matrices As in one dimension, computing the average of each measure of correlation will involve computing multiple 14 terms of the form tr � PE|ψ⟩⟨ψ|⊗k� , (93) where P has a similar tensor product structure as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (41), adapted to the two-dimensional setting con- sidered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The number of required replicas k again depends on the considered measure of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will thus need a two-dimensional analog to the transfer matrices introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In contrast to the one-dimensional case, the size of the resulting transfer matrices will also depend on the geometry of the consid- ered subsystems, making their analysis much more chal- lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, the procedure of defining the tensors is similar to that in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We provide an overview here and refer to App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' L for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We define V (i,j) = U (i,j) ⊗ U (i,j), where U (i,j) ∈ U � dD2� is the unitary matrix depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By computing the k-fold twirl [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (32)], we obtain the building block = � dU (94) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (95) As in one dimension, it is convenient to define a build- ing block for which the contraction of bond (blue) legs is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Analogously to the one-dimensional case, this results in a tensor with only permutation-valued (green) legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' L, the resulting bulk tensor is given via = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (96) For the sake of brevity, the expressions for the boundary tensors are stated in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Computing expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (93) corre- sponds to contracting each S with some P (d) ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The entries of the resulting bulk tensor Tρ ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' are given by = = (97) = � σ∈Sk Wg � στ −1, dD2� d#(σρ)D#(σθ−1)D#(σν−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (98) In our computations, the site in the top right corner belongs to neither A nor B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is thus acted upon by the trivial permutation e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' L, the corresponding tensor is given by Te = |Fk⟩⟨Fk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' L, we furthermore show that a property similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (55) also holds for random isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, tensors corresponding to the trivial permutation e ∈ Sk on the boundary simplify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For example, = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (99) As for the one-dimensional case, the proper normaliza- tion of E|ψ⟩⟨ψ|⊗k follows directly from this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Instead of thinking in terms of contractions of two- dimensional tensor networks, it will later prove beneficial to think again in terms of multiplications of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, for any height h of the subsystems A and B [see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a)], we define = (100) 15 as well as = and = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (101) For subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a), we can now write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (93) in terms of transfer matrices: tr � PE|ψ⟩⟨ψ|⊗k� = ⟨Ik|T c e T a α T r e T b β |Fk⟩ (102) We provide an additional Mathematica package [63] that defines Tρ with ρ ∈ Sk for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , 20} according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Once again, the package relies on the one provided by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [64] for evaluating the Weingarten function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Estimating the Decay of Correlations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (102) implies that, for subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a), the decay of the average of each measure of correlation will again reduce to a statement in terms of transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, the decay will be determined by the spectrum of Te with e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Notice that, in addition to a dependence on k, the form and properties of Te now depend also on the height h of the subsystems A and B [see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, as we prove in the following sections, its two leading eigen- values are independent of h for at least k = 2 and k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Crucially, this will allow us to make statements about the decay of the averages of I2(A : B) and N(A : B) for arbitrary h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Note that we could, in principle, investigate vertical separation instead of horizontal separation because = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (103) The underlying exchange of indices does not affect the spectrum of the relevant identity transfer matrix and thus neither the average correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This reflects the fact that the sequential generation procedure is symmet- ric in the horizontal and vertical spatial directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Rényi-2 Mutual Information In this section, we compute the average of the Rényi-2 mutual information I2(A : B) for random isoTNS that are generated as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Subsystems A and B are defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, we will make the assumption that E log(X) = log(EX) (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1 of the proof of Result 4 (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' O) is thus all but identical to Step 1 of the proof of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Also Step 2 is largely analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To see this, let us take E tr � ϱ2 A � as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With A and B as defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) and x = (12), E tr � ϱ2 A � = (104) = (105) = ⟨I2|T c e T a x |F2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (106) The resulting expression for EI2(A : B), EI2(A : B) = log � ⟨I2|T c e T a x T r e T b x |F2⟩ � − log � ⟨I2|T c e T a x |F2⟩ � − log � ⟨I2|T c+a+r e T b x |F2⟩ � , (107) resembles Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (64) closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The remaining technical challenge is the analysis of the spectrum of Te with e ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote its distinct eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' M that λ1 = 1 and λ2 = dD3 − dD d2D4 − 1 (108) and that both eigenvalues are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the proof, that holds for any h, we map the contraction of tensors defining Te with e ∈ S2 to a multiplication of ma- trices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the Weingarten calculus, we show that Te is upper block triangular, a property that simplifies the analysis of its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The main difficulty is then to prove that the specified λ2 is indeed the subleading eigen- value for all h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We do this by exploiting substochasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Following the same reasoning as before, we find that the average of I2(A : B) decays exponentially as specified in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We prove this result in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Result 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of I2(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ2D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Trace Distance and 2-Norm In this section, we show the exponential decay of the average of N(A : B) for random isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As for the one- dimensional case, we do that to eventually make conclu- sions about the behavior of the trace distance T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While it is not trivial to compute the average of N(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a), the computation follows along the lines of what we have laid out in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we find that the decay of the average of N(A : B) is determined by the spectrum of the transfer matrix Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' N, for any h, λ1 = 1 and λ2 = dD3 − dD d2D4 − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (109) As in one dimension, the former is non-degenerate, while the degeneracy of the latter is given by the number of transpositions in S4, w2 = �4 2 � = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (110) While the analysis of the spectrum is, in principle, similar to that of the spectrum of Te with e ∈ S2, it is consider- ably more technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This is largely due to the fact that the matrices whose multiplication defines Te with e ∈ S4 are significantly more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Still, we find also Te with e ∈ S4 to be upper block triangular, allowing us to show that the specified λ2 is indeed the subleading eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given λ2 of Te with e ∈ S4 and the arguments devel- oped in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV D, we can state the first result of this section, which we prove in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Result 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of N(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ2D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We now turn to the case of correlations in isoTNS with arbitrary (but fixed) subsystems A and B [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The decay of the average of N(A : B) for arbitrary A and B can be bounded by employing the fact that the Schatten 2-norm satisfies [70] ∥trB(XAB)∥2 ≤ � dim(B)∥XAB∥2, (111) where XAB is any bipartite operator and dim(B) is the dimension of the Hilbert space that is traced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This means that N(A : B) ≤ d|AC|+|BC|N(A′ : B′), (112) where A is now an arbitrary subsystem, A′ is its (min- imal) enclosing rectangle that touches the hypersurface, and AC = A′ − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B′ and BC are defined analogously for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the statement above, we can bound the decay of the average of N(A : B) for arbitrary A and B as a straightforward corollary of Result 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We stress that we consider regime in which the distance r between A and B grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For arbitrary subsystems A and B, the average of N(A : B) with respect to the random isoTNS ensemble decays as N(A : B) = O � exp � − r ξ2D �� , (113) where the average correlation length ξ2D is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we state a concentration result for N(A : B), which, in combination with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (16), also allows us to draw conclusions about the typical behavior of T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As before, we consider arbitrary (but fixed) subsystems A and B [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For arbitrary subsystems A and B, suffi- ciently large r, and all 0 < ε < 1, the random isoTNS ensemble satisfies Pr � N(A : B) ≥ K exp � −(1 − ε)r ξ2D �� ≤ exp � − εr ξ2D � , (114) where K is constant with respect to r and the average correlation length ξ2D is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For arbitrary subsystems A and B, suffi- ciently large r, and all 0 < ε < 1, the random isoTNS ensemble satisfies Pr � T(A : B) ≥ K exp � −(1 − ε)r 2ξ2D �� ≤ exp � − εr ξ2D � (115) where K is constant with respect to r and the average correlation length ξ2D is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The proofs and discussions of these results are identical to their one-dimensional counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The tools we have developed in this and the previous two sections should, in principle, allow us to make state- ments also about the decay of the average of the von Neumann mutual information I(A : B) with respect to the random isoTNS ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, we would need to investigate the spectrum of Te with e ∈ Sk for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, as the analysis of the spectrum of Te is already quite technical for e ∈ S4 with our methods, we refrain from tackling the spectrum for k ≥ 5 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' CONCLUSION We have investigated the average behavior of correla- tions between two distant subsystems A and B for en- sembles of random MPS and isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As measures of 17 correlation, we have considered the Rényi-α mutual in- formation, a measure arising from the Hilbert–Schmidt norm, the trace distance, and the von Neumann mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We have shown that the average of each considered measure exhibits an exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our results can equivalently be seen as describing states re- sulting from quantum circuits with a sequential architec- ture and Haar random gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By leveraging the Weingarten calculus, we have devel- oped a mathematical framework that allows to infer the average correlation length from the subleading eigenvalue of an appropriately defined transfer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We have computed the averages of the Rényi-α mutual informa- tion and the measure arising from the Hilbert–Schmidt norm to show the emergence of an average correlation length that only depends on the underlying spatial di- mension but not the considered measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, the average correlation length for random MPS increases weakly with the bond dimension D and converges rapidly (as D grows) to the value 1/ log(d), where d is the phys- ical dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' On the contrary, the average correlation length for random isoTNS, while still depending on the bond dimension only weakly, decreases with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Surpris- ingly, the highest average correlation length for random isoTNS is achieved with the lowest non-trivial bond di- mension (D = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using elementary concentration results, we have fur- thermore deduced the typical behavior of the measure arising from the Hilbert–Schmidt, which has in turn al- lowed us to make similar statements about the trace dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For MPS, we have been able to give strong indications that the universal correlation length applies also to the von Neumann mutual information, and also any Rényi- α mutual information for integer values of α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It would be interesting to prove this behavior rigorously, and also investigate its validity for the isoTNS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' An- other possible future direction would be to study average correlations in more general random PEPS, beyond the class of isoTNS, as well as other types of quantum circuit architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Ignacio Cirac and Philippe Faist for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' and G.' metadata={'source': 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Lat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 14, 631 (2017), arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='04493 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [77] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Akers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Faulkner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Rath, Reflected entropy in random tensor networks, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2022, 162, arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='09122 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [78] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Biane, Some properties of crossings and partitions, Discrete Mathematics 175, 41 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [79] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Castellani and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Cavagna, Spin-glass theory for pedestrians, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2005, P05012 (2005), arXiv:cond-mat/0505032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix A: k-Fold Twirl In this appendix, we present some additional details on the k-fold twirl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we go from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (18) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To do this, we will need a result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [54], which appears as Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='4 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We state it without proof as Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let k be a positive integer, and (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , ik), (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , jk), (ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , ℓk), and (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , mk) be k-tuples of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, � dU Ui1ji · · · UikjkUm1ℓ1 · · · Umkℓk = � σ,τ∈Sk Wg � σ−1τ, q � δi1mσ(1) · · · δikmσ(k)δj1ℓτ(1) · · · δjklτ(k), (A1) where the integration is with respect to the Haar measure on the unitary group U(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 20 By Lemma 1, � T (k) U (X) � i1···ikm1···mk = � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',jk ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',ℓk � dU Ui1ji · · · UikjkXj1···jkℓ1···ℓkUm1ℓ1 · · · Umkℓk (A2) = � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',jk ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',ℓk � σ,τ∈Sk Wg � σ−1τ, q � δi1mσ(1) · · · δikmσ(k)Xj1···jkℓ1···ℓkδj1ℓτ(1) · · · δjklτ(k) (A3) = � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',jk � σ,τ∈Sk Wg � σ−1τ, q � δi1mσ(1) · · · δikmσ(k)Xj1···jkjτ−1(1)···jτ−1(k) (A4) = � σ,τ∈Sk Wg � σ−1τ, q �� P (q) σ−1 � i1···ikm1···mk tr � XP (q) τ � , (A5) where, in the final line, we have used that � P (q) σ−1 � i1···ikm1···mk = ⟨i1 · · · ik|P (q) σ−1|m1 · · · mk⟩ (A6) = ⟨i1 · · · ik|mσ(1) · · · mσ(k)⟩ (A7) = δi1mσ(1) · · · δikmσ(k) (A8) and that tr � XP (q) τ � = � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',jk ⟨j1 · · · jk|XP (q) τ |j1 · · · jk⟩ (A9) = � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',jk ⟨j1 · · · jk|X|jτ −1(1) · · · jτ −1(k)⟩ (A10) = Xj1···jkjτ−1(1)···jτ−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (A11) Thus, T (k) U (X) = � σ,τ∈Sk Wg � σ−1τ, q � P (q) σ−1 tr � XP (q) τ � (A12) = � σ,τ∈Sk Wg � στ −1, q � P (q) σ tr � XP (q) τ −1 � (A13) = � σ,τ∈Sk Wg � στ −1, q � P (q) σ tr � X � P (q) τ �T � , (A14) which coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In addition, let us confirm that T (k) U � P (q) ρ � = P (q) ρ [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Indeed, T (k) U � P (q) ρ � = � σ,τ∈Sk Wg � στ −1, q � P (q) σ tr � P (q) ρ � P (q) τ �T � (A15) = � σ,τ∈Sk Wg � στ −1, q � P (q) σ q#(ρτ −1) (A16) = � σ∈Sk δρσP (q) σ (A17) = P (q) ρ , (A18) where, in the third line, we have used the definition of the Weingarten function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix B: Proofs of Propositions 1 and 2 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The eigenvalues of Te with e ∈ Sk are non-negative for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 21 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Te with e ∈ Sk is diagonalizable for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To prove Propositions 1 and 2, we will define matrices X ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' and Y ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' so that Ck = WXY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will then discuss some properties of those three matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The proofs themselves will boil down to similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We define the diagonal matrix X ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' via Xσσ = d#(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (B1) It is evident that X is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We define the Gram matrix Y ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' via Yσθ = tr � P (D) σ � P (D) θ �T � = D#(σθ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (B2) Y is positive semidefinite because it is a Gram matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The Weingarten matrix Wg � στ −1, q � = � G−1� στ is positive definite because the Gram matrix G [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (21)] is positive definite [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will need the fact that Y W is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Y W has non-negative eigenvalues because it is a product of a positive semidefinite (Y ) and a positive definite matrix (W) (see Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [73]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Furthermore, Y W is symmetric: ⟨τ|Y W|θ⟩ = � σ∈Sk Wg � στ −1, dD � D#(σθ−1) (B3) = � π∈Sk Wg � πθτ −1, dD � D#(π) (B4) = � π∈Sk Wg � τθ−1π−1, dD � D#(π) (B5) = � π∈Sk Wg � θ−1π−1τ, dD � D#(π) (B6) = � ϕ∈Sk Wg � θ−1ϕ, dD � D#(τϕ−1) (B7) = � ϕ∈Sk Wg � ϕθ−1, dD � D#(ϕτ −1) (B8) = ⟨θ|Y W|τ⟩ (B9) In the third line, we have used that Wg(α, q) = Wg � α−1, q � , and, in the fourth line, we have used that Wg(α, q) = Wg � βαβ−1, q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Both identities are a result of the Weingarten function being sensitive only to the conjugacy class of a given permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Ck = WXY is similar to XY W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A product of a positive definite (X) and a positive semidef- inite matrix (Y W), XY W has non-negative eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows by similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Ck = WXY is similar to XY W, which is similar to X1/2Y WX1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because Y W is symmetric, so is X1/2Y WX1/2, which makes the latter diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix C: Proof of Proposition 3 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Te with e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, λ1 = 1 and it is non-degenerate for any k ≥ 2 if d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For what follows, it is convenient to define the transfer matrix Σe with e ∈ Sk via = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C1) 22 Our strategy will be to prove the claimed spectral property for Σe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This is enough because, for any two operators X and Y for which XY and Y X are well defined, the sets of eigenvalues of XY and Y X are the same (up to zeros and the multiplicity of the non-zero eigenvalues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (51) and (C1), Tρ and Σρ are related in exactly this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As Σe arises from the contraction of the quantum channel underlying R [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (44)] with the identity permutation, it can be understood as a generalization of the k-fold twirling operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, it involves an ”environment” E of dimension � Cd�⊗k that is eventually traced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As a superoperator (that is, without using the operator-vector correspondence), it reads [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (18)] Σe(·) = � dU trE � U ⊗k � k � l=1 |0⟩⟨0|El ⊗ (·)Sl � � U †�⊗k � , (C2) where the integration is with respect to the Haar measure on the unitary group U(dD), E = �k l=1 El corresponds to � Cd�⊗k, and S = �k l=1 Sl corresponds to � CD�⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is apparent that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C2) represents a convex combination of quantum channels (note the Stinespring dilation form), and thus the resulting operator is also a valid quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This implies that 1 is an eigenvalue of Σe and that there is no eigenvalue of greater modulus [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We now prove that no other eigenvalue of the same modulus exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, we will show that Σe is a primitive channel [74], which implies said property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A quantum channel is primitive if and only if the spanning space formed by products of its Kraus operators, Km = span �� m � k=1 Kik �� , (C3) is equal to the full matrix algebra for some integer m, that is, Km = MDk(C) [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show that this condition is satisfied for Σe if d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Indeed, the Haar integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C2), together with the partial trace, can be understood as a (redundant) Kraus decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Precisely, we can take {Ki}i = � trE �� k � l=1 |+⟩⟨ψl|El ⊗ ISl � U ⊗k �� U,ψ , (C4) where U ∈ U(dD), |ψl⟩ ∈ {|0⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , |d − 1⟩} is the computational basis of the lth replica of the environment, and |+⟩ = �d−1 j=0 |j⟩/ √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The latter is a choice (instead of |0⟩) made for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It remains to show that there exists an integer m = m(k, d, D) such that Km = MDk(C) if d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' First of all, note that the above fails for d = 1 (that is, if the environment E is trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This is because span �� U ⊗k� U � coincides with the symmetric subspace over the k subsystems [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, d = 2 is already enough to span the full matrix algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' An explicit construction to show this fact amounts to taking U to be a controlled unitary gate, where the control system is E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, consider |ψl⟩ = |δlr⟩ and U = |0⟩⟨0|E ⊗ IS + d−1 � j=1 |j⟩⟨j|E ⊗ VS, (C5) where r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k}, and VS is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This results in Kraus operators [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C4)] of the form IS1 ⊗ · · · ⊗ ISr−1 ⊗ V ⊗ ISr+1 · · · ⊗ ISk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C6) Taking (finite) products, as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C3) dictates, is enough to build a basis of the vector space MDk(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Note that this construction requires two control levels (that is, d ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix D: Further Properties of the Transfer Matrix Te In this appendix, we state and prove statements about the structure of the transfer matrix Te with e ∈ Sk that will motivate Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We prove the statements in Apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' E, F, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Moving forward, we denote Te with e ∈ Sk by Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Each entry ⟨τ|Ck|θ⟩ = � σ∈Sk Wg � στ −1, dD � d#(σ)D#(σθ−1) (D1) 23 of Ck ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' is a sum of k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While this may sound daunting, Ck exhibits a structure that reduces its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As formalized by Proposition 4, the entries ⟨τ|Ck|θ⟩ of Ck exhibit a dependence on the conjugacy class of their indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, if we know the entries of the column given by θ ∈ Sk, we also know the entries of the columns given by permutations in the same conjugacy class as θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any π ∈ Sk, ⟨τ|Ck|θ⟩ = ⟨πτπ−1|Ck|πθπ−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D2) Certain entries of Ck vanish, while others are given by the entries of Ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 5 captures these two statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For all θ ∈ Sk with θ(k) = k, ⟨τ|Ck|θ⟩ = δk,τ(k)⟨τ ↓|Ck−1|θ↓⟩, (D3) where ρ↓ ∈ Sk−1 is the restriction of ρ ∈ Sk with ρ(k) = k to the permutation on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To understand the strength of Propositions 4 and 5, let us have a look at C2 and C3, the two most simple transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With bases S2 = {e, (12)} and S3 = {e, (12), (13), (23), (123), (132)}, one finds that C2 = � 1 α 0 β � and C3 = � � � � � � � 1 α α α γ γ 0 β 0 0 δ δ 0 0 β 0 δ δ 0 0 0 β δ δ 0 0 0 0 ε ζ 0 0 0 0 ζ ε � � � � � � � , (D4) where each Greek letter corresponds to some function of d and D whose exact form is not relevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The entries in the first four columns of C3 are fully determined by those of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The entries in the last two columns of C3 do not arise from those of C2, but the sixth column is a permutation of the fifth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Note that we are deliberately choosing a basis Sk = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , sk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='} that makes the special structure of Ck more apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we sort permutations so that those with i fixed points come before those with i − 1 fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We group permutations that have common fixed points and then those that are in the the same conjugacy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given this basis, Propositions 4 and 5 imply that Ck is block triangular with k diagonal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We denote by C(1) k the diagonal block with τ = θ = e ∈ Sk and by C(i) k with 2 ≤ i ≤ k the diagonal block corresponding to τ, θ ∈ Sk with k − i fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The spectrum of Ck is then given by λ(Ck) = λ � C(1) k � ∪ λ � C(2) k � ∪ · · · ∪ λ � C(k) k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D5) It is apparent that C(1) k has a single, non-degenerate eigenvalue µ1 = ⟨e|Ck|e⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D6) C(2) k has a single, degenerate eigenvalue µ2 = ⟨(12)|Ck|(12)⟩ = dD2 − d d2D2 − 1, (D7) which corresponds to the expression of β in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The degeneracy of µ2 is given by the size of the block, which is in turn given by the number of transpositions in Sk, v2 = �k 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D8) By Proposition 3, 1 is the leading eigenvalue of Ck and non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We conjecture that µ2 is the subleading eigenvalue of Ck and that it has degeneracy v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement of Conjecture 1 holds for k ∈ {2, 3, 4}, and we have numerical evidence suggesting so for k ∈ {5, 6, 7} [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 24 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, λ2 = µ2 with degeneracy w2 = v2 for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As formalized by Proposition 6, the statements above hold for any transfer matrix Tρ with ρ ∈ Sk because Tρ is similar to Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any ρ ∈ Sk, Tρ = QT ρ CkQρ with Qρ = � π∈Sk |ρπ⟩⟨π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D9) Appendix E: Proof of Proposition 4 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any π ∈ Sk, ⟨τ|Ck|θ⟩ = ⟨πτπ−1|Ck|πθπ−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that ⟨τ|Ck|θ⟩ = � σ∈Sk Wg � στ −1, dD � d#(σ)D#(σθ−1) (E1) = � ϕ∈Sk Wg � π−1ϕπτ −1, dD � d#(π−1ϕπ)D#(π−1ϕπθ−1) (E2) = � ϕ∈Sk Wg � ϕπτ −1π−1, dD � d#(π−1ϕπ)D#(ϕπθ−1π−1) (E3) = � ϕ∈Sk Wg � ϕ � πτπ−1�−1, dD � d#(ϕ)D # � ϕ(πθπ−1) −1� (E4) = ⟨πτπ−1|Ck|πθπ−1⟩, (E5) where, in the second line, we have used that the conjugation map is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the third line, we have used that Wg(α, q) = Wg � βαβ−1, q � and that #(α) = # � βαβ−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Both identities are a result of the two functions being sensitive only to the conjugacy class of a given permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix F: Proof of Proposition 5 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For all θ ∈ Sk with θ(k) = k, ⟨τ|Ck|θ⟩ = δk,τ(k)⟨τ ↓|Ck−1|θ↓⟩, (D3) where ρ↓ ∈ Sk−1 is the restriction of ρ ∈ Sk with ρ(k) = k to the permutation on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To prove Proposition 5, we will need a number of ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The main one will be Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [76], which we state without proof as Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any ρ ∈ Sk, k−1 � i=1 Wg � (ik)π, q � + q Wg(π, q) = δk,π(k) Wg � π↓, q � , (F1) where (ik) denotes the transposition of elements i and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To use Lemma 2, we must do two things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' First, we must split the sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D1) into certain sums of k terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We employ Lemma 3 to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any α ∈ Sk, there exists a β ∈ Sk with β(k) = k such that either α = β or α = (ik)β with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If α(k) = k, then α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Otherwise, k is in exactly one of the disjoint cycles of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Without loss of generality, say α = (kia3a4 · · · )(b1b2 · · · ) · · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, α = (ik)(ia3a4 · · · )(b1b2 · · · ) · · · ≡ (ik)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Second, we must ensure that summands in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D1) with Wg(π, dD) have a factor dD while those with Wg � (ik)π, dD � do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We achieve this with Lemma 5 whose proof employs Lemma 4 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [77], which we state without proof as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The lemma also appears as Lemma 1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let α ∈ Sk be a transposition, and β ∈ Sk such that #(β) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If the elements exchanged by α are not in the same cycle of β, then #(αβ) = u − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let ϕ, θ ∈ Sk with ϕ(k) = k and θ(k) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' # � ϕθ−1� = # � (ik)ϕθ−1� + 1 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' # � ϕθ−1� − #(ϕ) = # � (ik)ϕθ−1� − # � (ik)ϕ � for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let ϕ, θ ∈ Sk with ϕ(k) = k and θ(k) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, � ϕθ−1� (k) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, k is in a cycle by itself and thus not in the same cycle of ϕθ−1 as i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let ϕ, θ ∈ Sk with ϕ(k) = k and θ(k) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, i and k are not in the same cycle of neither ϕθ−1 nor ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By Lemma 4, # � ϕθ−1� = # � (ik)ϕθ−1� + 1 and #(ϕ) = # � (ik)ϕ � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, # � ϕθ−1� − #(ϕ) = # � (ik)ϕθ−1� + 1 − # � (ik)ϕ � − 1 = � (ik)ϕθ−1� − # � (ik)ϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With that, we can prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By Lemma 3, ⟨τ|Ck|θ⟩ = � σ∈Sk Wg � στ −1, dD � d#(σ)D#(σθ−1) (F2) = � ϕ∈Sk ϕ(k)=k �k−1 � i=1 Wg � (ik)ϕτ −1, dD � d# � (ik)ϕ � D#((ik)ϕθ−1) + Wg � ϕτ −1, dD � d#(ϕ)D#(ϕθ−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (F3) By Lemma 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' for θ ∈ Sk with θ(k) = k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ⟨τ|Ck|θ⟩ = � ϕ∈Sk ϕ(k)=k 1 d#(ϕθ−1)−#(ϕ) �k−1 � i=1 Wg � (ik)ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � (dD)#((ik)ϕθ−1) + Wg � ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � (dD)#(ϕθ−1) � (F4) = � ϕ∈Sk ϕ(k)=k (dD)#(ϕθ−1)−1 d#(ϕθ−1)−#(ϕ) �k−1 � i=1 Wg � (ik)ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � + dD Wg � ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � � (F5) = � ϕ∈Sk ϕ(k)=k d#(ϕ)−1D#(ϕθ−1)−1 �k−1 � i=1 Wg � (ik)ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � + dD Wg � ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � � (F6) = � ϕ∈Sk ϕ(k)=k d#(ϕ↓)D # � (ϕθ−1] ↓��k−1 � i=1 Wg � (ik)ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � + dD Wg � ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (F7) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' in the final line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' we have used that # � α↓� = #(α) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By Lemma 2, ⟨τ|Ck|θ⟩ = � ϕ∈Sk ϕ(k)=k δk,(ϕτ −1)(k) Wg �� ϕτ −1�↓, dD � d#(ϕ↓)D # � (ϕθ−1) ↓� (F8) = δk,τ(k) � ϕ∈Sk ϕ(k)=k Wg �� ϕτ −1�↓, dD � d#(ϕ↓)D # � (ϕθ−1) ↓� (F9) = δk,τ(k)⟨τ ↓|Ck−1|θ↓⟩, (F10) 26 where, in the second line, we have used that � ϕτ −1� (k) = k if and only if τ(k) = k because ϕ(k) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix G: Proof of Proposition 6 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any ρ ∈ Sk, Tρ = QT ρ CkQρ with Qρ = � π∈Sk |ρπ⟩⟨π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that ⟨τ|Tρ|θ⟩ = � σ∈Sk Wg � στ −1, dD � d#(σρ)D#(σθ−1) (G1) = � σ∈Sk Wg � ρστ −1ρ−1, dD � d#(ρσ)D#(ρσθ−1ρ−1) (G2) = � σ∈Sk Wg � ρσ(ρτ)−1, dD � d#(ρσ)D#[ρσ(ρθ)−1] (G3) = � π∈Sk Wg � π(ρτ)−1, dD � d#(π)D#[π(ρθ)−1] (G4) = ⟨ρτ|Ck|ρθ⟩, (G5) where, in the second line, we have used that Wg(α, q) = Wg � βαβ−1, q � and that #(α) = # � βαβ−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Both identities are a result of the two functions being sensitive only to the conjugacy class of a given permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the fourth line, we have used that the left-multiplication map is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix H: Proof of Result 2 Result 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of N(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We split the proof into four steps, following the structure of the proof of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We rewrite EN(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With the Hilbert-Schmidt inner product, EN(A : B) = E tr � ϱ2 AB � + E tr � ϱ2 A � tr � ϱ2 B � − 2E tr[ϱAB(ϱA ⊗ ϱB)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(H1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='It is then easy to confirm that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='EN(A : B) = tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='P (d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='�⊗c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⊗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='P (d) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We express EN(A : B) in terms of the transfer matrices defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given the previous step, it is easy to confirm that EN(A : B) = ⟨I4|T c e T a (12)T r e T b (12)|F4⟩ + ⟨I4|T c e T a (34)T r e T b (12)|F4⟩ − 2⟨I4|T c e T a (12)T r e T b (13)|F4⟩ (H3) = ⟨I4|T c e T a (12)T r e � T b (12) + T b (34) − 2T b (13) � |F4⟩ (H4) ≡ ⟨I4|T c e AT r e B|F4⟩, (H5) where we have defined A = T a (12) and B = T b (12) + T b (34) − 2T b (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H6) 27 Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EN(A : B) in terms of the spectrum of Te with e ∈ S4 [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (58)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds [63] that λ1 = 1 and λ2 = dD2 − d d2D2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H7) The former is non-degenerate, while the degeneracy of the latter is given by the number of transposition in S4 [63], w2 = �4 2 � = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H8) Expanding T c e and taking the limit c → ∞ yields EN(A : B) = ⟨L1|AT r e B|F4⟩, (H9) where we have used that ⟨I4|R1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' After expanding T r e and using that |F4⟩ = |R1⟩, we have EN(A : B) = ⟨L1|A|R1⟩⟨L1|B|R1⟩ + λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ + O(λr 3) (H10) = λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ + O(λr 3), (H11) where, in the second line, we have used that ⟨L1|B|R1⟩ = 0, (H12) which we prove in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we prove that ⟨L1|T b t |R1⟩ does not depend on the two elements the transposition t ∈ S4 acts upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We need some understanding of T b t as well as the eigenvectors ⟨L1| and |R1⟩ of Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For the former, we make use of the fact that T b ρ with ρ ∈ S4 is similar to to T b e with e ∈ S4, T b ρ = � π,ϕ∈S4 |π⟩⟨ρπ|Cb 4|ρϕ⟩⟨ϕ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H13) With α = d2D − D d2D2 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' β = dD2 − d d2D2 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' f(u) = u−1 � i=0 αβi and g(u) = βu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(H14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='Te|si⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1≤i≤7 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H15) For getting some understanding of ⟨L1|, we make use of the fact that ⟨L(µ) i |R(ν) j ⟩ = δijδµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is easy to confirm that |R1⟩ = |s1⟩ and |R(µ) 2 ⟩ = α β − 1|s1⟩ + |sµ+1⟩, (H16) which implies that � ⟨L1|si⟩ � 1≤i≤7 = � 1 α β − 1 α β − 1 α β − 1 α β − 1 α β − 1 α β − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H17) For any transposition t ∈ S4, it thus holds that ⟨L1|T b t |R1⟩ = � π,ϕ∈S4 ⟨L1|π⟩⟨tπ|Cb 4|tϕ⟩⟨ϕ|R1⟩ (H18) = � π,ϕ∈S4 ⟨L1|π⟩⟨tπ|Cb 4|tϕ⟩δs1ϕ (H19) = � π∈S4 ⟨L1|π⟩⟨tπ|Cb 4|t⟩ (H20) = � π∈S4 f(b)⟨L1|π⟩⟨tπ|s1⟩ + � π∈S4 g(b)⟨L1|π⟩⟨tπ|t⟩ (H21) = � π∈S4 f(b)⟨L1|π⟩δtπ + � π∈S4 g(b)⟨L1|π⟩δs1π (H22) = f(b)⟨L1|t⟩ + g(b)⟨L1|s1⟩ (H23) = α β − 1f(b) + g(b), (H24) which is independent of the two elements the transposition t ∈ S4 acts upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, ⟨L1|B|R1⟩ = ⟨L1| � T b (12) + T b (34) − 2T b (13) � |R1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H25) 29 Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we can write EN(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, EN(A : B) ≡ K exp � −r ξ � + O � exp � − r χ �� , (H26) where K = w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ (H27) and ξ = − 1 log(λ2) = − � log � dD2 − d d2D2 − 1 ��−1 = ξ1D > χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H28) This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix I: Proof of Result 3 and Corollary 4 Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, the average of I(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, for any integer value of α ≥ 1, the average of Iα(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In this appendix, we prove Result 3 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The proof of the latter will follow directly from the proof of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We split the proof into four steps, following the structure of the proof of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because I(A : B) and I2(A : B) are related, the steps are overall very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As is usual in the context of the replica trick [79], we will interchange the order of some limits without rigorous justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We rewrite EI(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, we make use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (88) and (89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With those, EI(A : B) = lim α→1 lim v→0 1 vα − v � log � E tr(ϱα AB)v� − log � E tr(ϱα A)v� − log � E tr(ϱα B)v�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I1) E tr(ϱα A)v, E tr(ϱα A)v, and E tr(ϱα AB)v can be written in the desired form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let us define x ∈ Sα to be the cyclic permutation so that x(i) = i + 1 modulo α and xw = � α(w − 1) + 1, α(w − 1) + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , αw � ∈ Svα (I2) with w ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, for example, E tr(ϱα AB)v = tr ��� P (d) e �⊗c ⊗ � P (d) x �⊗a ⊗ � P (d) e �⊗r ⊗ � P (d) x �⊗b ⊗ � P (d) e �⊗(f−1) ⊗ P (dD) e � E|ψ⟩⟨ψ|⊗α �v (I3) = tr ��� P (d) e �⊗c ⊗ � P (d) x1···xv �⊗a ⊗ � P (d) e �⊗r ⊗ � P (d) x1···xv �⊗b ⊗ � P (d) e �⊗(f−1) ⊗ P (dD) e � E|ψ⟩⟨ψ|⊗vα � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I4) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We express EI(A : B) in terms of the transfer matrices defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given the previous step, it is easy to confirm that EI(A : B) = lim α→1 lim v→0 1 vα − v � log � ⟨Ivα|T c e T a x1···xvT r e T b x1···xv|Fvα⟩ � − log � ⟨Ivα|T c e T a x1···xv|Fvα⟩ � − log � ⟨Ivα|T c+a+r e T b x1···xv|Fvα⟩ �� (I5) ≡ lim α→1 lim v→0 1 vα − v � log(⟨Ivα|T c e AT r e B|Fvα⟩) − log(⟨Ivα|T c e A|Fvα⟩) − log � ⟨Ivα|T c+a+r e B|Fvα⟩ �� (I6) where we have defined A = T a x1···xv and B = T b x1···xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I7) 30 Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EI(A : B) in terms of the spectrum of Te with e ∈ Svα [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (58)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' At this point, we assume Conjecture 1 to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, for any vα ≥ 2, we assume that λ2 = dD2 − d d2D2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I8) with degeneracy w2 = �k 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I9) Expanding T c e and taking the limit c → ∞ yields EI(A : B) = lim α→1 lim v→0 1 vα − v [log(⟨L1|AT r e B|Fvα⟩) − log(⟨L1|A|Fvα⟩) − log(⟨L1|B|Fvα⟩)], (I10) where we have used that ⟨Ivα|R1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' After expanding T r e and using that |Fvα⟩ = |R1⟩, we have EI(A : B) = lim α→1 lim v→0 1 vα − v � log � ⟨L1|A|R1⟩⟨L1|B|R1⟩ + λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ + O(λr 3) � − log(⟨L1|A|R1⟩) − log(⟨L1|B|R1⟩) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I11) Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we can write EI(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With Λ = max �� λ2r 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' λr 3 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' EI(A : B) = lim α→1 lim v→0 1 vα − v log � 1 + λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ ⟨L1|A|R1⟩⟨L1|B|R1⟩ + O(λr 3) � (I12) = lim α→1 lim v→0 1 vα − v � λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ ⟨L1|A|R1⟩⟨L1|B|R1⟩ + O(Λ) � (I13) ≡ lim α→1 lim v→0 1 vα − v � �K(vα) exp � −r ξ � + O � exp � − r χ ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I14) where �K(vα) = w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ ⟨L1|A|R1⟩⟨L1|B|R1⟩ (I15) and ξ = − 1 log(λ2) = − � log � dD2 − d d2D2 − 1 ��−1 = ξ1D > χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I16) As ξ is independent of vα, it cannot be affected by the replica limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' �K(vα) will converge to some K that is guaranteed to be independent of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (89), it holds that EIα(A : B) = lim v→0 1 vα − v � log � E tr(ϱα AB)v� − log � E tr(ϱα A)v� − log � E tr(ϱα B)v�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I17) Thus, the proof is identical to that of Result 3 without the limit α → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 31 Appendix J: Result 3 with c = 0 In this appendix, we prove a version of Result 3 with c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Steps 1 and 2 of this proof are identical to Steps 1 and 2 of the proof of Result 3 with c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, at the end of Step 2, we have EI(A : B) = lim α→1 lim v→0 1 vα − v � log(⟨Ivα|ACr vαB|Fvα⟩) − log(⟨Ivα|A|Fvα⟩) − log � ⟨Ivα|Ca+r vα B|Fvα⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (J1) We start the proof at Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EI(A : B) in terms of the spectrum of Te with e ∈ Svα [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (58)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We assume Conjecture 1 to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Expanding T r e and using that ⟨Ivα|R1⟩ = 1 yields EI(A : B) = lim α→1 lim v→0 1 vα − v � log � ⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ + λr 2 w2 � µ=1 ⟨Ivα|A|R(µ) 2 ⟩⟨L(µ) 2 |B|Fvα⟩ + O(λr 3) � − log(⟨Ivα|A|Fvα⟩) − log � ⟨L1|B|Fvα⟩ + λa+r 2 w2 � µ=1 ⟨Ivα|R(µ) 2 ⟩⟨L(µ) 2 |B|Fvα⟩ + O(λr 3) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (J2) Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We write EI(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With Λ = max �� λ2r 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' λr 3 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='EI(A : B) = lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='α→1 lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='v→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='vα − v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 + λr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='µ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='+ O(λr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='− log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 + λa+r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='µ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨L1|B|Fvα⟩ + O(λr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(J3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='= lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='α→1 lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='v→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='vα − v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='λr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='µ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='+ λa+r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='µ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨L1|B|Fvα⟩ + O(Λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(J4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='= lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='α→1 lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='v→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='vα − v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='λr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='µ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='+ λa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2⟨Ivα|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨L1|B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='+ O(Λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(J5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='≡ lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='α→1 lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='v→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='vα − v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='K′(vα) exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='− r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='χ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (J6) where � K′(vα) = w2 � µ=1 � ⟨Ivα|A|R(µ) 2 ⟩⟨L(µ) 2 |B|Fvα⟩ ⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ + λa 2⟨Ivα|R(µ) 2 ⟩⟨L(µ) 2 |B|Fvα⟩ ⟨L1|B|Fvα⟩ � (J7) and ξ = − 1 log(λ2) = − � log � dD2 − d d2D2 − 1 ��−1 ξ1D > χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (J8) Again, � K′(vα) will converge to some K′ that is guaranteed to be independent of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While K′ is different from K in general, the correlation length ξ is independent of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix K: Numerical Analysis In this appendix, we briefly review our numerical analysis of the von Neumann mutual information I(A : B) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We fix d and D, and we set a = b = 1 and r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (i) We generate a + r + b + 1 Haar-random unitary matrices of U(dD) to define |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This definition makes the assumption that there are no sites before subsystem A (that is, c = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We discuss in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' J why this does not affect the average correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By setting f = 1, we furthermore use the fact that the sites after subsystem B do not play a role as a result of the sequential generation [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (55)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (ii) We compute I(A : B) with respect to |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (iii) We repeat steps (i) and (ii) 10 000 times to compute the average of I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (iv) We repeat steps (i) through (iii) for r ∈ {7, 9, 11, 13, 15}, plot the averages of I(A : B) against r, and fit the data to extract the average correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (v) To obtain Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 3, we repeat steps (i) through (iv) for different d and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 32 Appendix L: Transfer Matrices in Two Dimensions This appendix expands on Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will state the definitions of the boundary tensors and prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From the main text, recall that we define V (i,j) = U (i,j) ⊗ U (i,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By computing the k-fold twirl, we obtain the building block = � dU = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (L1) With that, we have E|ψ⟩⟨ψ|⊗k = = , (L2) where, in the final step, we have cut permutation-valued (green) legs instead of bond (blues) ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' S is the tensor stated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' S′ and S′′ reflect the different boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Instead of first stating the tensors with bond (blue) legs, let us immediately state those with permutation-valued (green) legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The tensors at the top boundary are given via = = (L3) = � σ∈Sk Wg � στ −1, dD2� (dD)#(σρ)D#(σθ−1), (L4) and those at the right boundary are given via = = (L5) = � σ∈Sk Wg � στ −1, dD2� (dD)#(σρ)D#(σν−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (L6) 33 We will always contract S′′ with P (d) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The tensor in the top-right corner thus plays the same role as the final vector |Fk⟩ = e1 ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' does in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In fact, Te = |Fk⟩⟨Fk|, = = ≡ (L7) = � σ∈Sk Wg � στ −1, dD2�� dD2�#(σe) = δeτ, (L8) where we have defined = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (L9) If a tensor corresponding to e ∈ Sk is contracted with |Fk⟩, it factorizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For tensors at the top boundary, we have = = = (L10) = � σ∈Sk Wg � στ −1, dD2�� dD2�#(σe) = δeτ, (L11) for those at the right boundary, we have = = = (L12) = � σ∈Sk Wg � στ −1, dD2�� dD2�#(σe) = δeτ, (L13) and for those in the bulk, we have = = = (L14) = � σ∈Sk Wg � στ −1, dD2�� dD2�#(σe) = δeτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (L15) 34 The identities above lead to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (99): = = = (L16) = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (L17) Appendix M: Spectrum of the Transfer Matrix Te with e ∈ S2 In this appendix, we state and prove two lemmas concerning the spectrum of Te with e ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let us start with some preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We define |0⟩ = � 1 0 � , |1⟩ = � 0 1 � , and |+⟩ = � 1 1 � , (M1) and map the contraction of tensors defining Te with e ∈ S2 to a multiplication of matrices: = = , (M2) where = � � � 1 α α γ 0 0 0 0 0 0 0 0 0 β β δ � � � (M3) with α = d2D3 − D d2D4 − 1 , β = dD3 − dD d2D4 − 1 , γ = d2D2 − D2 d2D4 − 1 , and δ = dD2 − d d2D4 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M4) Note that = 0 if i ̸= j, (M5) 35 and N = = � 1 α 0 β � (M6) is equal to Te with e ∈ S2 for d → dD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With that, it easy to check that = = ⟨o|N|i⟩⟨i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M7) We also introduce an analytical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We define Mj = I⊗(j−1) ⊗ M ⊗ I⊗(h−j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M8) Te with e ∈ S2 is then given by Te = � I⊗h ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗h� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M9) With i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , ih ∈ {0, 1} and o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , oh ∈ {0, 1}, the entries of Te with e ∈ S2 are given by � ⟨o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , oh| ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ |i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , ih⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M10) Our first lemma states that Te with e ∈ S2 is block triangular, where our definition of blocks arises from the indexing of rows and columns in base 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, with 2 ≤ j ≤ h, the jth diagonal block of Te, which we denote by T (j) e ∈ R2j−1×2j−1, has fixed indices ih = · · · ij+1 = oh = · · · = oj+1 = 0 and ij = oj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M14), it is given by T (j) e = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M11) The first diagonal block, which we denote by T (1) e ∈ R2×2, is given by T (1) e = = = � 1 α 0 β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M12) In particular, we will prove that a block of Te is zero if its defining row digit oj is higher than its defining column digit ij, which implies that Te is upper block triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As the proof relies exclusively on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M7), Te inherits its upper block triangularity from the upper triangularity of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Te with e ∈ S2 is upper block triangular because � I⊗(j−1) ⊗ ⟨0| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ ⊗ |0⟩⊗(h−j)� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M13) 36 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From = |0⟩ (M14) and = 0, (M15) it follows that = = 0, (M16) which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In our second lemma, we utilize the block triangularity of Te with e ∈ S2 to make a direct statement about its two leading eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let |λ1| > |λ2| > · · · ≥ 0 denote the distinct eigenvalues of Te with e ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, for any h, λ1 = 1 and λ2 = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Furthermore, λ1 and λ2 are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The spectrum of Te with e ∈ S2 is given by the union of the spectra of its diagonal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is evident that the first block T (1) e has eigenvalues 1 and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the following, we show that any other diagonal block T (j) e with 2 ≤ j ≤ h can be written as a product of β and a strictly substochastic matrix, implying that its eigenvalues are strictly smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We structure the proof in steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show that any diagonal block T (j) e with 2 ≤ j ≤ h can be written as β times a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From = β|1⟩, (M17) it follows that T (j) e = = β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M18) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We argue that the matrix (M19) 37 is strictly column substochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that = � � � 1 α α γ 0 0 0 0 0 0 0 0 0 β β δ � � � (M20) is column substochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While the first column of M evidently sums to 1, the sums of the other columns are strictly bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As a result, Mj−1 · · · M1 is column substochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The boundary condition |1⟩ does not affect this because it specifies a subset of columns of the matrix Mj−1 · · · M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In fact, it imposes strict substochasticity because this subset does not include the only column summing to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Also the boundary condition ⟨+| does not affect the substochasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The boundary condition means that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M19) is a sum of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Each of these matrices comprises a disjoint subset of rows of the matrix Mj−1 · · · M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because Mj−1 · · · M1, the matrix comprising the whole set of rows, is column substochastic, so is the sum of the matrices comprising the disjoint subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 and β are the only eigenvalues of T (1) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' They are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because the matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M20) is strictly column substochastic, the eigenvalues of any diagonal block T (j) e with 2 ≤ j ≤ h are strictly smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As a preparation for App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' N, we provide the proofs of Lemmas 6 and 7 in analytical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Lemma 6 in analytical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From � ⟨0| ⊗ ⟨0| � M � I ⊗ |0⟩ � = |0⟩ (M21) and � ⟨0| ⊗ ⟨0| � M � I ⊗ |1⟩ � = 0, (M22) it follows that � I⊗(j−1) ⊗ ⟨0| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ ⊗ |0⟩⊗(h−j)� = � I⊗(j−1) ⊗ ⟨0| ⊗ ⟨0| �� Mj · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ � (M23) = 0, (M24) which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Lemma 7 in analytical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in the version with graphical notation, we structure the proof in steps, without repeating the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From � ⟨1| ⊗ ⟨0| � M � I ⊗ |0⟩ � = β|1⟩, (M25) it follows that T (j) e = � I⊗(j−1) ⊗ ⟨1| ⊗ ⟨0| �� Mj · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ � (M26) = β � I⊗(j−1) ⊗ ⟨1| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M27) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that � I⊗(j−1) ⊗ ⟨1| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� (M28) is strictly column substochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 and β are the only eigenvalues of T (1) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' They are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because the matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M28) is strictly column substochastic, the eigenvalues of any diagonal block T (j) e with 2 ≤ j ≤ h are strictly smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 38 Appendix N: Spectrum of the Transfer Matrix Te with e ∈ S4 In this appendix, we state and prove two lemmas concerning the spectrum of Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the same notation as in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' M, we will draw on results from that appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Te with e ∈ S4 is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M2), now with M ∈ R576×576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' it holds that � ⟨o| ⊗ ⟨0| � M � I ⊗ |i⟩ � = ⟨o|N|i⟩⟨i|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(N1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='N = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨+| ⊗ I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='I ⊗ |0⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='= ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 α α α α α α γ γ γ γ γ γ γ γ η η η η η η ρ ρ ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 β 0 0 0 0 0 δ δ δ δ 0 0 0 0 θ θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι θ σ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 β 0 0 0 0 δ δ 0 0 δ δ 0 0 ι θ θ θ θ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(N2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='is equal to Te with e ∈ S4 for d → dD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' M, our first lemma states that Te with e ∈ S4 is block triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The definition of blocks now arises from the indexing of rows and columns in base 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, any diagonal block (but the first) is defined by ih = · · · = ij+1 = oh = · · · = oj+1 = 0 and ij = oj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' There are four classes of diagonal blocks: The first diagonal block is in its own class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is given by � I ⊗ ⟨0|⊗(h−1) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I ⊗ |0⟩⊗(h−1)� = � I ⊗ ⟨0| � M � |+⟩ ⊗ I � = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N3) The second class of diagonal blocks corresponds to transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ij and oj correspond to the same transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' There are six subblocks in this class because there are six different transpositions in S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The third class of diagonal blocks corresponds to permutations with a single fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ij and oj correspond to any of the two permutations with the same single fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' There are four subblocks in this class because there are four different choices of a single fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The fourth class of diagonal blocks corresponds to permutations with no fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ij and oj correspond to any of the nine permutations with no fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we will prove that a block of Te is zero if its defining row digit oj is higher than its defining column digit ij stand in a certain relation to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As the proof relies exclusively on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N1), Te again inherits its upper block triangularity from the upper triangularity of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Te with e ∈ S4 is upper block triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 39 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N1), it follows that � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� = 0 (N4) if ij corresponds to the trivial permutation and oj does not, ij corresponds to a transposition and oj corresponds to a different transposition or a permutation with one or no fixed point, ij corresponds to a permutation with a single fixed point and oj corresponds to a permutation with a different single fixed point or no fixed point, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In our second lemma, we utilize the block triangularity of Te with e ∈ S4 to make direct a statement about its two leading eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let |λ1| > |λ2| > · · · ≥ 0 denote the distinct eigenvalues of Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, for any h, λ1 = 1 and λ2 = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Furthermore, λ1 is non-degenerate, and λ2 has a degeneracy of six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As is the case for Te with e ∈ S2, the spectrum of Te with e ∈ S4 is given by the union of the spectra of its diagonal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The two leading eigenvalues of the first diagonal block N are 1 and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the following, we show that any other diagonal block can be written as a product of β and a matrix whose spectral radius is strictly bounded by 1, implying that its eigenvalues are strictly smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We again structure the proof in steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show that any diagonal block but the first can be written as a product of beta and a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' it follows that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' if ij and oj correspond to the same transposition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� = � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0| �� Mj · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ � (N5) = β � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N6) if ij and oj correspond to any two permutations with the same single fixed point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� = � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0| �� Mj · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ � (N7) = p � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� (N8) < β 2 � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N9) where {ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ζ} ∋ p < β/2 [63] depends on ij and oj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' if ij and oj correspond to any permutation with no fixed point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� = � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0| �� Mj · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ � (N10) = p � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� (N11) < β 9 � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N12) where {µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' χ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ψ} ∋ p < β/9 [63] depends on ij and oj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 40 Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We now argue that the spectral radius of the matrix � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� (N13) is strictly bounded by 1 for 2 ≤ j ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that the spectral radius of M is bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While the first column of |M| sums to 1, the sums of the other columns are strictly bounded by 1 [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As a result, the spectral radius of Mj−1 · · · M1 is bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The boundary condition |o⟩ does not affect this because it specifies a subset of columns of the matrix Mj−1 · · · M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In fact, it imposes a strict bound because this subset does not include the only column summing to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Also the boundary condition ⟨+| does not affect the bound on the spectral radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The boundary condition means that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N13) is a sum of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Each of these matrices comprises a disjoint subset of rows of the matrix Mj−1 · · · M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because the spectral radius of Mj−1 · · · M1, the matrix comprising the whole set of rows, is bounded by 1, so is the sum of the matrices comprising the disjoint subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 and β are the two leading eigenvalues of the first diagonal block N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' They are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because the spectral radius of the matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N13) is strictly bounded by 1, the eigenvalues of any other diagonal block are strictly smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix O: Proof of Result 4 Result 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of I2(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ2D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We split the proof into four steps, following the structure of the proof of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The steps are overall very similar to those of that proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We rewrite EI2(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, we make the assumption that E log(X) = log(EX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, EI2(A : B) = log � E tr � ϱ2 AB �� − log � E tr � ϱ2 A �� − log � Etr � ϱ2 B �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (O1) E tr � ϱ2 A � , E tr � ϱ2 B � , and E tr � ϱ2 AB � can be written in the desired form [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (62) and (63)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We express EI2(A : B) in terms of the transfer tensors defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V A and use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (100) to map contractions of two-dimensional tensor networks to multiplications of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The latter is enabled by our definition of subsystems A and B [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We have done this for E tr � ϱ2 A � in graphical notation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V C [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (104) and (105)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is easy to confirm that EI2(A : B) = log � ⟨I2|T c e T a (12)T r e T b (12)|F2⟩ � − log � ⟨I2|T c e T a (12)|F2⟩ � − log � ⟨I2|T c+a+r e T b (12)|F2⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (O2) Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EI2(A : B) in terms of the spectrum of Te with e ∈ S2, which we consider in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we know λ1 and λ2 as well as their algebraic and geometric multiplicities, we do not need Te to be diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Expanding T c e and taking the limit c → ∞ yields EI2(A : B) = log � ⟨L1|T a (12)T r e T b (12)|F2⟩ � − log � ⟨L1|T a (12)|F2⟩ � − log � ⟨L1|T b (12)|F2⟩ � , (O3) where we have used that ⟨I2|R1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' After expanding T r e and using that |F2⟩ = |R1⟩, we have I2(A : B) = log � ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ + λr 2⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ + O � rv−1λr 3 �� − log � ⟨L1|T a (12)|R1⟩ � − log � ⟨L1|T b (12)|R1⟩ � , (O4) where v denote the size of the largest Jordan block with respect to λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 41 Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we can write EI2(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With Λ = max �� λ2r 2 , rv−1λr 3 �� , EI2(A : B) = log � 1 + λr 2 ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ + O � rv−1λr 3 � � (O5) = λr 2 ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ + O(Λ) (O6) ≡ K exp � −r ξ � + O � exp � − r χ �� , (O7) where K = ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ (O8) and ξ = − 1 log(λ2) = − � log �dD3 − dD d2D4 − 1 ��−1 = ξ2D > χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (O9) This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix P: Proof of Result 5 Result 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of N(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ2D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We split the proof into four steps, following the structure of the proof of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The steps are overall very similar to those of the proof of Result 2 (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We rewrite EN(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, with the Hilbert-Schmidt inner product, EN(A : B) = E tr � ϱ2 AB � + E tr � ϱ2 A � tr � ϱ2 B � − 2E tr[ϱAB(ϱA ⊗ ϱB)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P1) The right-hand side can be written in the desired form [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We express EN(A : B) in terms of the transfer tensors defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V A and use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (100) to map contractions of two-dimensional tensor networks to multiplications of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The latter is enabled by our definition of subsystems A and B [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is easy to confirm that EN(A : B) = ⟨I4|T c e T a (12)T r e T b (12)|F4⟩ + ⟨I4|T c e T a (34)T r e T b (12)|F4⟩ − 2⟨I4|T c e T a (12)T r e T b (13)|F4⟩ (P2) = ⟨I4|T c e T a (12)T r e � T b (12) + T b (34) − 2T b (13) � |F4⟩ (P3) ≡ ⟨I4|T c e AT r e B|F4⟩, (P4) where we have defined A = T a (12) and B = T b (12) + T b (34) − 2T b (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P5) Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EN(A : B) in terms of the spectrum of Te with e ∈ S4, which we consider in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we know λ1 and λ2 as well as their algebraic and geometric multiplicities, we do not need Te to be diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Expanding T c e and taking the limit c → ∞ yields EN(A : B) = ⟨L1|AT r e B|F4⟩, (P6) 42 where we have used that ⟨I4|R1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' After expanding T r e and using that |F4⟩ = |R1⟩, we have EN(A : B) = ⟨L1|A|R1⟩⟨L1|B|R1⟩ + λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ + O(λr 3) (P7) = λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ + O(λr 3), (P8) where, in the second line, we have used that ⟨L1|B|R1⟩ = 0, (P9) which we prove in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in the proof of Result 2 (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' H), we prove that ⟨L1|T b t |R1⟩ does not depend on the two elements the transposition t ∈ S4 acts upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In fact, the proof follows from the proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H12) of that appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We just need two additional considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' First, the proof of Lemma 8 is not specific to the trivial permutation e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, Tρ exhibits an upper block triangular structure for any ρ ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The first diagonal block is given by Tρ with ρ ∈ S4, � ⟨si|Tρ|sj⟩ � 1≤i≤24 1≤j≤24 = Tρ, (P10) which implies that � ⟨si|T b ρ |sj⟩ � 1≤i≤24 1≤j≤24 = T b ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P11) Second, the eigenvectors ⟨L1| and |R1⟩ of Te with e ∈ S4 arise from those of Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, |R⟩1 = s1 and � ⟨L1|si⟩ � 1≤i≤7 = � 1 α β − 1 α β − 1 α β − 1 α β − 1 α β − 1 α β − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P12) ⟨L1|T b t |R1⟩ thus does not depend on the two elements the transposition t ∈ S4 acts upon because ⟨L1|T b t |R1⟩ does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This implies that ⟨L1|B|R1⟩ = ⟨L1| � T b (12) + T b (34) − 2T b (13) � |R1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P13) Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we can write EN(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, EN(A : B) ≡ K exp � −r ξ � + O � exp � − r χ �� , (P14) where K = w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ (P15) and ξ = − 1 log(λ2) = − � log �dD3 − dD d2D4 − 1 ��−1 = ξ2D > χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P16) This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} diff --git a/1dFKT4oBgHgl3EQfOy1w/vector_store/index.faiss 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b/4tAyT4oBgHgl3EQfQPYV/content/tmp_files/2301.00039v1.pdf.txt @@ -0,0 +1,981 @@ +New Insights on the Stokes Paradox for Flow in +Unbounded Domains +Ingeborg G. Gjerde and L. Ridgway Scott +January 3, 2023 +Abstract +Stokes flow equations, used to model creeping flow, are a commonly used simplifi- +cation of the Navier–Stokes equations. The simplification is valid for flows where the +inertial forces are negligible compared to the viscous forces. In infinite domains, this +simplification leads to a fundamental paradox. +In this work we review the Stokes paradox and present new insights related to +recent research. We approach the paradox from three different points of view: modern +functional analysis, numerical simulations, and classical analytic techniques. The first +approach yields a novel, rigorous derivation of the paradox. We also show that relaxing +the Stokes no-slip condition (by introducing a Navier’s friction condition) in one case +resolves the Stokes paradox but gives rise to d’Alembert’s paradox. +The Stokes paradox has previously been resolved by Oseen, who showed that it +is caused by a limited validity of Stokes’ approximation. We show that the paradox +still holds for the Reynolds–Orr equations describing kinetic energy flow instability, +meaning that flow instability steadily increases with domain size. We refer to this as +an instability paradox. +1 +Introduction +Fluids like air and water can be modeled to high precision using the Navier–Stokes equa- +tions [29]. These equations are still intensively studied, and they have some very curious +mathematical features, often described as paradoxes, such as d’Alembert’s paradox [10, 26] +and Whitehead’s paradox [30, section 8.3]. Here we discuss one of the most well known, +namely the Stokes paradox. The Stokes paradox also arises for more exotic fluids, such as +Fermi electrons [17]. +Consider a domain Ω ⊂ R3 containing an infinitely long cylinder with radius 1. The +cylinder has a boundary denoted Γ. The domain is filled with a fluid moving with velocity +u and pressure p past the cylinder. Mathematically, this can be described using the Navier– +Stokes equations +−∆u + ∇p = −R u · ∇u +in Ω, +∇· u = 0 +in Ω, +(1) +together with the boundary condition +u = g on Γ. +(2) +1 +arXiv:2301.00039v1 [physics.flu-dyn] 30 Dec 2022 + +In this form, the Navier–Stokes equations are posed with only one parameter, namely +the Reynolds number R [15]: +R = UL/ν. +(3) +Here, U is the representative flow velocity, L the representative length for the domain, and +ν is the kinematic fluid viscosity. +In this work, we are interested in the case where the Reynolds number will be small, and +we have R ≪ 1. In this case, it seems reasonable to drop the advective term, in which case +we have the simpler Stokes equations +−∆u + ∇p = 0 in Ω, +∇· u = 0 in Ω, +(4) +again with a boundary condition of the form (2). +If the domain Ω is taken to be infinitely large, however, this simplification leads to a +fundamental paradox [27, 6]. The Stokes paradox is that “creeping flow of an incompressible +Newtonian fluid around a cylinder in an unbounded fluid has no solution” [28]. Others have +characterized the paradox by saying +• [23, 1st paragraph] “Stokes (1851) established that there was no solution to the two- +dimensional, steady, incompressible, Navier–Stokes equations for asymptotically uni- +form flow around a cylinder.” +• [1, Remark 3.15] “no classical solution ... that tends to a nonzero vector at infinity.” +These statements require some clarification. Firstly, potential flow around a cylinder solves +the Stokes equations with the right boundary conditions at infinity. However, potential flow +(Figure 1a) does not satisfy no-slip boundary conditions on the cylinder; for this reason it +has commonly been discarded as physically incorrect. +To handle the no-slip boundary condition we instead look to the literature on functional +analysis. A precise mathematical theory for the Stokes equations in an unbounded domain +was developed in [9]. We can explain the results there informally as follows: That it is +a well posed problem to require a reasonable flow profile around a reasonable domain. A +solution therefore exists solving Stokes equations in an infinite domain enclosing a cylinder, +as long as we set reasonable boundary conditions on the cylinder. But the paradox is that +we cannot set a boundary condition at infinity. +This point is often misunderstood, saying instead that there is a “need to satisfy two +boundary conditions, one on the object and one at infinity” [4]. This is despite the fact +that many papers have dealt with the paradox and its resolution. The first resolution was +by Oseen [19] who realized that it was necessary to keep R > 0 in (1) and provided an +approximate (linearized) solution to Navier-Stokes. This observation has been substantially +amplified by Finn [6]. But many others have dealt with the paradox, for example by improv- +ing the Oseen approximation [14, 20]. For a survey and history of their methods, see [30, +Chapter V] where the so-called matched asymptotic expansions are traced back to seminal +work by K. O. Friedrichs. +The Sobolev space in which the solution exists provides important context for the Stokes +paradox. To be more precise, the solution exists in a special weighted Sobolev space in which +we give up control over the function as we get infinitely far from the cylinder. For this reason +it is not possible to prescribe the flow at infinity. We will show that, if the flow is specified +2 + +to be a constant on the cylinder, the flow must be that constant everywhere. Thus, while a +non-trivial solution exists, it may not be physically meaningful. +Due to the confusion related to the Stokes paradox, we examine it in some detail before +proceeding to the main results of the paper. We draw together disparate approaches to give +the fullest possible understanding. Although much of what we present at the beginning can +be found separately elsewhere, the combination of the different approaches gives a more +complete picture than has so far been presented in one place. +The article will proceed as follows. We begin in Section 2 by giving two analytic solutions +for Stokes flow around a cylinder and discuss their properties. Next, we give in Section 3 +a derivation of the Stokes paradox, using the mathematical framework from [9]. In Section +4, we examine the impact of a different boundary condition on the cylinder, Navier’s slip +(friction) condition. We see that it is possible to resolve the Stokes paradox with one value +of the friction parameter, although that value seems to be nonphysical. +We next view the Stokes paradox through other lenses. In particular, we solve the Stokes +problem on a very large (but finite) domain surrounding the cylinder. In this case we can +pose any boundary conditions we like on the outer boundary. So what goes wrong as we let +the outer boundary go to infinity? In Section 5 we answer this in two ways, using modern +computational methods (Section 5.1) and classical analytical solution techniques (Section +5.2). We will see that they give the same answer: the solution goes to a constant. Thus +we have multiple ways of viewing the Stokes paradox, each with its own advantages and +limitations. +In Section 6 we review the resolution of the Stokes paradox by Oseen [19]. According +to Oseen, the paradox is caused by the limited validity of Stokes’ approximation, which +relies on the Reynolds number being small. To be more precise, we have two limits going to +infinity: the viscosity and the domain size. Either limit alone is well behaved, but jointly +they are not. Thus one must consider the full Navier-Stokes equations if the domain is +large. We also discuss briefly some open questions regarding the existence of a solution for +the Navier–Stokes equations when the Reynolds number grows large. +In Section 7 we describe extensions of the Stokes paradox concepts to other flow problems. +In particular, we discuss how the Stokes paradox relates to flow instabilities. The Reynolds- +Orr method gives a way to calculate the kinetic energy instability of a perturbed base +flow, where the most unstable flow perturbations are calculated by solving a Stokes type +eigenvalue problem. In [11], it was observed that the instability of a base flow kept increasing +as the domain grew. This can be explained as a particular instance of the Stokes paradox. +To the best of our knowledge, however, it cannot be resolved in any such way as Oseen’s +resolution of the Stokes paradox. +2 +Classical solutions +Assume for the moment Γ to be an infinitely long cylinder of radius 1 in the z-direction with +origin (0, 0) in the x, y-plane. Consider now the Stokes flow equations (4) with boundary +condition (2). We now construct two analytic solutions. The strategy is to find a function +u that is divergence free so it satisfies the second equation in (4). If the Laplacian of u is a +gradient of some function, we can solve the first equation in (4) by construction by taking +the pressure equal to said function. +To find a function u that is divergence free, we use two different approaches. For the +first, we take φ to be a potential function, i.e. we set u = ∇φ = (φx, φy), where φi denotes +3 + +(a) Potential flow, β = −2ν +(b) Zero friction flow, β = 0 +Figure 1: Stream function ψ, stream lines and flux u (cones) for two classically known +analytic solutions of the Stokes equations for flow past an infinitely long cylinder. Neither +solution satisfies the no-slip boundary condition. +the derivative of φ in the ith direction. Then if we want ∇·u = 0 we want ∇·∇φ = ∆φ = 0, +i.e. φ to be harmonic. Many harmonic functions are known in the literature; we take +φ(r, θ) = −(cos θ) +� +r − 1 +r +� += −x +� +1 − 1 +r2 +� +. +(5) +A straightforward calculation shows that φ is harmonic. +We also have +∆u = (∆φx, ∆φy) = ((∆φ)x, (∆φ)y) = 0. +Thus we have a solution to the Stokes equation (4) if the pressure is taken to be a constant +(so that ∇p = 0 as well). +Differentiating r = +� +x2 + y2 we find +rx = x +r , +ry = y +r . +(6) +Using these, we have +φx = 1 − r−2 + 2yr−3ry = 1 + (y2 − x2)r−4, +φy = 2yr−3rx = 2xyr−4. +Then +u(x, y) = (φx(x, y), φy(x, y)) = +� +1 + +y2 − x2 +(x2 + y2)2 , +−2xy +(x2 + y2)2 +� +. +(7) +This solution for u is commonly referred to as potential flow. The potential flow solution +is shown in Figure 1a together with its stream function and streamlines. We see that u goes +to uniform flow at infinity. Moreover, u · n = 0 on the cylinder. However, the tangential +4 + +u +2 +1.5 +0.5 +0 +7.5 +4 +2 +0 +-2 +-4 +-7.5n +2 +1.5 +0.5 +0 +20 +10 +0 +-10 +-20velocity u · τ ̸= 0, meaning that this solution does not satisfy a no-slip boundary condition +on Γ. This led Stokes to reject this solution. +Let us therefore make another analytic solution, this time trying the second approach +to making a divergence-free flux u. By augmenting our vectors with a z-component we +can define u to be the curl of a vector (0, 0, ψ) so that u = (ψy, −ψx, 0). Dropping the +z-coordinate again we have u = (ψy, −ψx) and ∇· u = ψyx − ψxy = 0 as long as ψ is +sufficiently smooth1. For example, define +ψ = (sin θ) r log r = y log r. +(8) +Then +u(x, y) = +�y2 +r2 + log(r), −xy +r2 +� +. +(9) +By construction u satisfies the second equation in (4). The first equation in (4) can +then be satisfied by choosing p so that ∇p = ∆u. We will show in Section 5.2 that such a +solution p exists. +Again, the solution does not satisfy the no-slip boundary condition on Γ. Moreover, the +solution diverges to infinity as r → ∞. +In the next section, we see how these analytic solutions are related to the Stokes paradox. +3 +Derivation of Stokes paradox using the variational +framework +The Stokes paradox occurs for incompressible flow Stokes flow with no-slip boundary condi- +tions on the cylinder (i.e. g = 0 in (2)). The paradox can be derived in several ways. In this +work, we present three approaches: (i) a rigorous derivation using weighted Sobolev spaces, +(ii) a formal approach using simulations in domains of increasing size and (iii) a semi-formal +approach of deriving analytic solutions in bounded domains and passing to the limit. +3.1 +Sobolev spaces for the Stokes equations +If the domain Ω is Lipschitz continuous [8, section 5.1]), the system is well posed with +u ∈ (H1(Ω))d, d ∈ {2, 3} being the dimension of the domain, and p ∈ L2(Ω/R), where +L2(Ω)/R = +� +v : Ω → R : +� +Ω +v2 dx < ∞, +� +Ω +v dx = 0 +� +is the space of square-integrable functions, together with the norm given by +∥v∥L2(Ω) = +�� +Ω +v2 dx. +We also define +H1(Ω) = +� +v ∈ L2(Ω) : ∇v ∈ L2(Ω) +� +1for example ψ ∈ C2(Ω), i.e. two times continuously differentiable. In this case we can change the order +of differentiation without issues. +5 + +and the norm +∥v∥H1(Ω) = +�� +Ω +|∇v(x)|2 + v(x)2 dx, +where |v(x)| is the Euclidean norm of ∇v(x). +The notation u ∈ (H1(Ω))d then means +that every component of u is in H1(Ω). To simplify notation, we from now on drop the +superscript and simply write u ∈ H1(Ω). +In summary, given a bounded domain and reasonable f and g (for example f ∈ L2(Ω) +and g ∈ H1(Ω) [8]) there exists a unique solution pair u ∈ H1(Ω) and p ∈ L2(Ω) of (4). +Once we know there exists a unique solution, we can use numerical methods to solve for its +approximation. +If the domain Ω is infinite, the previous result no longer holds. +Instead, the Stokes +equation will be well posed with p ∈ L2(Ω)/R and u ∈ H1 +w(Ω) defined by the norm +∥v∥H1w(Ω) = +�� +Ω +|∇v(x)|2 + +� +1 + |x| log |x| +�−2|v(x)|2 dx. +(10) +Centrally, this is a weaker norm than the one we had for the H1(Ω). Both spaces require the +gradient of the function itself to be square-integrable, but in the H1 +w(Ω)-space we only require +the function to be square-integrable when multiplied by a weight function +� +1+|x| log |x| +�−1. +As this weight function goes to zero as x → ∞, the function does not have to decay at all +as we move away from the cylinder. As we will see it may even diverge. Therefore we have +to be very careful about assigning limit values at infinity to functions u ∈ H1 +w(Ω). +What can be said, based on [9], is that one cannot specify boundary conditions simulta- +neously on the cylinder and at infinity. That is, having specified conditions on the cylinder, +the conditions at infinity have already become specified. We will see that the resolution of +the Stokes paradox is simple once we know in what function spaces to look for solutions. +We now know that solutions exists for the Stokes problem, but only in a certain weighted +Sobolev space. Due to the weight going to zero as we move away from the cylinder, the +solution is allowed to behave more mischievously in this region. +In the time of Stokes +(1819–1903), the functional analysis approach to partial differential equations was still in +its infancy. Indeed, it was not until 1991 [9] that the appropriate function spaces were fully +clarified. +3.2 +Derivation of the Stokes paradox +Now that we know there exists a solution, we can straightforwardly formulate the Stokes +paradox. For this, it is useful to choose moving coordinates. Instead of thinking of a fixed +cylinder in a moving fluid, let us reverse the point of view by using moving coordinates such +that the fluid appears at rest. If we think of a moving cylinder in an infinite fluid, we can pose +the (Navier–)Stokes equations as in (4) with a boundary function g = (1, 0), assuming the +cylinder is moving in the x-direction with unit speed. In view of [9], there is a unique solution +u ∈ H1 +w(Ω), where Ω is the complement of the cylinder (i.e. {(x, y) ∈ R2 : x2 + y2 > 1}) +and fixed in time. +But g ∈ H1 +w(Ω), and g is a solution of (4), with constant pressure. And [9] proves that +g is the solution. Thus we have proved the following theorem. +Theorem 3.1 Suppose that we move an infinite cylinder in a direction perpendicular to the +axis of the cylinder with unit speed. If the entirety of the fluid is governed by the Stokes +6 + +equations (4) with a no-slip boundary condition on the cylinder, then the entirety of the fluid +is forced to move at unit speed. +In the variational framework, the Stokes paradox is not really a paradox: The Stokes +equation is well posed in infinite domains, but the appropriate function space for u is one +where we give up control of u as it approaches infinity. Thus it is not surprising that the +solution is non-physical away from the cylinder. +The fact that we are not able to specify the limiting value of the solution raises the +question of how badly the solution might behave. We investigate this in the next section. +3.2.1 +Limit values of functions in H1 +w +In the previous section we saw that moving an infinitely long cylinder through an infinite +domain Ω with given speed g = (1, 0) caused the entire solution flux to be u = (1, 0). In +fact, due to the weight function, any v ∈ H1 +w(Ω) can tend to a nonzero constant at infinity. +Worse, it can grow like (log r)α for α < 1/2, as long as its gradient remains square integrable. +In particular, take u(r) = (log r)α. Then for r > 1, +|∇u| = +���α∇r +r (log r)α−1��� = +���α x +r2 (log r)α−1��� = +���α +r (log r)α−1���. +This expression is square integrable at infinity if +� ∞ +K +r dr +r2(log r)2(1−α) < ∞. +Changing coordinates to s = log r (so that r−1dr = ds), our condition reduces to +� ∞ +log K +ds +s2(1−α) < ∞. +This holds when α < 1/2. +Thus the Stokes equation posed in an infinite domain may have a solution that diverges +as |x| → ∞. An example of this is the zero friction solution (9), as we now discuss. +4 +Navier’s revenge +In Section 3.2 we saw how the imposition of a no-slip boundary condition on the cylinder +leads to the Stokes paradox in unbounded domains. In this section we will discuss what +may happen for different boundary conditions. We will see that the Stokes paradox does +not occur for all boundary conditions. +Instead of the no-slip boundary condition (2) with g = 0, let us consider Navier’s slip +condition. This boundary condition, sometimes referred to as Navier’s friction condition +[18, 12, 5], links the tangential velocity and the shear stress on Γ: +β u · τ k = −ν nt(∇u + ∇ut)τ k, +k = 1, 2, +(11) +where τ i are orthogonal tangent vectors and β is the friction coefficient. This is coupled +with the no-penetration condition u·n = 0 on Γ. In our two-dimensional case of flow around +a cylinder, there is only one tangent vector τ. The other one is perpendicular to the plane +7 + +of the two-dimensional flow, that is, parallel to the cylinder axis. For β > 0, the Navier +slip condition works as a friction causing the fluid to slow down as it slips over the cylinder +boundary Γ. +The potential function φ and stream function ψ defined in (5) and (8) give rise to +two different solutions of the Stokes equations with Navier’s boundary condition (11), each +corresponding to a particular choice of β. The potential flow solution (7), i.e. +u = (φx, φy) +⇒ +u(x, y) = +� +1 + +y2 − x2 +(x2 + y2)2 , +−2xy +(x2 + y2)2 +� +, +solves the Navier–Stokes equations for all ν, and satisfies the Navier slip condition if β = −2ν +[11]. Moreover, this flow goes to the desired asymptotic limit (zero) at infinity. Thus (7) +resolves Stokes’ paradox for β = −2ν. For this particular boundary condition the solution +belongs to the standard Sobolev space H1(Ω) and exhibits reasonable physical behavior in +the entire domain. +This raises the question of what happens for other values of β. Interestingly, the other +analytic solution we have, i.e. (9), satisfies the friction boundary condition (11) if β = 0. To +see this, note that τ = (−y, x) and n = (−x, −y) on Γ. Computing (∇u) τ we find +(∇u)τ = τ · ∇u = ∂θ +� +sin2 θ + log(r), − cos θ sin θ +� += +� +2 sin θ cos θ, − cos2 θ + sin2 θ +� +. +(12) +Therefore (omitting some trigonometric simplifications) +nt(∇u)τ|Γ = −(cos θ, sin θ)t� +2 sin θ cos θ, − cos2 θ + sin2 θ +� += −2 sin θ cos2 θ + cos2 θ sin θ − sin3 θ = − sin θ. +(13) +Similarly +(∇u)n = n · ∇u = −r∂r +� +sin2 θ + log(r), − cos θ sin θ +� += +� +− 1, 0 +� +. +(14) +This says that +τ t(∇u)n|Γ = (− sin θ, cos θ) · +� +− 1, 0 +� += sin θ. +(15) +Note that nt(∇ut)τ = τ t(∇u)n. Therefore +nt(∇u + ∇ut)τ|Γ = +� +nt(∇u)τ + τ t(∇u)n +� +|Γ = 0. +(16) +Thus the function in (9) solves the Stokes equations and satisfies the Navier slip condition +if β = 0. But this solution diverges as r → ∞, fast enough that the norm in (10) is not +finite. Thus (9) does not resolve the Stokes paradox. +It is worth noting that the fact that β = 0 does not mean that the drag on the cylinder is +zero [10]. It is known that the drag IS zero for β = −2ν, and this is the core of d’Alembert’s +paradox [10]. +The computations above are sufficiently complex that it is useful to have a way to verify +them. This can be done by solving the Stokes equations with the Navier slip condition +numerically with β = 0 and check that the result approximately agrees with (9). +For β > 0, the Navier slip condition acts as a friction force, slowing the flow as it slips +over the cylinder. For β → ∞, the Navier friction boundary condition converges to the +Stokes no-slip condition. For other values of β, the techniques in section 5 could be used to +see if there are plausible solutions of the Stokes paradox. +8 + +In conclusion, we see that the Stokes paradox is resolved using the Navier slip boundary +condition with one particular value for β, but not for others. Navier died in 1836, so he +was not available to comment on Stokes’ paradox. We can only wonder what he might have +said. +5 +Stokes flow on bounded domains of increasing size +In Section 3.1 we saw how the Stokes problem lost the ability to specify the value of the +solution on the boundary away from the cylinder. With this in mind, we now restrict our +attention to bounded domains, where it is possible to pose boundary conditions. We explore +two approaches, a computational one and an analytical one. We will see that as we increase +the size of the box, we again encounter the Stokes paradox. +5.1 +Computational approach +Recent advances in software [2, 13] have made it easy to solve partial differential equations +(PDEs). Using such software, you can study PDEs without knowing detailed background +prerequisites [21]. We now indicate this approach for the Navier–Stokes equations. +Consider the domain Ωb defined by +Ωb = {x : |x| > 1, |xi| < b, i = 1, 2} +(17) +for b > 1. Let Γ denote the subset of ∂Ωb defined by +Γ = {x : |x| = 1} , +that is, Γ represents the cylinder. +We keep our viewpoint of a cylinder moving through the larger domain where the fluid +is at rest. I.e., we consider solutions ub of the problem (4) with boundary conditions +ub = (1, 0) on Γ, +ub = 0 on ∂Ωb\Γ. +(18) +Figure 2 shows the solution for b = 4. +Figure 3 shows the horizontal component of the solution for (a) b = 16 and (b) b = 32. +We see that the support of the horizontal component spreads as the box gets bigger. Thus +we see that the horizontal component of the solutions is not really going to zero at the +boundary of the box. It remains positive as we go to the edge of the domain both upstream +and downstream of the cylinder. +To examine how the support of the horizontal component of the solution spreads as b is +increased, we considered a functional to examine the size of ub in regions of increasing size +d, but fixed independent of b. Thus we defined +� +Ωb +χd(x)2|ub(x)|2 dx +� � +Ωb +χd(x)2 dx +(19) +where χd(x) is the interpolant on the computational mesh of the cut-off function +1 +2 +� +1 − tanh +� +20 +� +|x|2 − d2��� +9 + +Figure 2: Plot of pressure p, flux u (glyphs) and streamlines for the solution the moving +cylinder problem (4) in the domain (17) with boundary conditions (18) for b = 7.5. Due to +no-slip boundary condition u = (0, 0) on the box walls, the fluid is forced to recirculate. +(a) +(b) +(c) +Figure 3: Plot of the horizontal component of the solution of (4) in the domain (17) with +boundary conditions (18) for (a) b = 16, M=64 and (b) b = 32, M=128. M is the mesh +parameter for mshr, with the number of segments for the definition of the circle chosen to +be M as well. +10 + +p +1.85 +0 +-1.85 +u +0.5 +0n +0.5 +0 +-0.25n +0.5 +0 +-0.250.75 +0.5 +0.25 +-32 +-16 +0 +16 +3 +-uo(x) for b=32 +-uo(x) for b=16which is very close to 1 inside |x| < d and very close to zero outside of that. If ub → (1, 0) +as b → ∞, then we would expect the expression (19) to increase to 1. If on the other hand, +if ub → 0 as r → ∞, we would expect the expression (19) to converge to some value less +than 1 as b is increased. +Figure 4 gives the data for three values of d as a function of box size b. It appears +(19) indeed increases to 1, which points to ub → (1, 0) for |x| < d as b → ∞. This is in +accordance with the Stokes paradox as stated in Theorem 3.1; that as b → ∞, we have +u = (1, 0) everywhere. But then the fluid moves like a solid. +Interestingly, u = (1, 0) satisfies the Navier slip condition with β = 0 since ∇u = 0. +Thus this solution not only satisfies the no-slip boundary condition on the cylinder, but +also the Navier friction condition with β = 0. The other solution with β = 0, i.e. (9) has +different boundary values on the cylinder. It also diverges when r → ∞, unlike the solution +u = (1, 0). +(a) +(b) +Figure 4: Growth of (19) as a function of r (horizontal axis) for three values of d: (top) +d = 10, (middle) d = 20, (bottom) d = 30. (a) Stokes no-slip boundary condition, (b) +Navier friction boundary condition, β = 0. +5.2 +Analytic solutions in bounded domains +Let us return from modern numerical software back to the classics. +In this section, we +consider analytical solutions in increasingly large circular domains, following [23]. These +domains are related to the so-called Leray approximate solutions [3]. +Following [28, (12)], consider a general biharmonic stream function of the form +ψ = f(r) sin θ, +f(r) = Ar−1 + Br log r + Cr3 + Dr. +Now let us show that the fact that ψ is biharmonic implies that u = (ψy, −ψx) satisfies +the first equation in (4). For the sake of calculations, let us for the moment augment the +domain with a z-component and let ψ = (0, 0, ψ) so that we can define u = curl ψ. Note +that ∇ · ψ = 0 since ψ depends only on x and y. By using the vector calculus identity +curl (curl v) = ∇(∇ · v) − ∆v, we then see +curl ∆u = curl (∆(curl ψ)) = curl curl ∆ψ = ∇ (∇ · ∆ψ) +� +�� +� +=0 +−∆2ψ. +11 + +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +104 +102 +1030.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +102 +103 +104Since all four terms in ψ are biharmonic in any open set that excludes the origin, we can +then conclude that u satisfies the following: curl ∆u = −∆2ψ = 0 in any open set that +excludes the origin. +Invoking Stokes’ theorem [8, Theorem 2.9], we conclude that ∆u = ∇p for some scalar +function p. Thus u satisfies (4). Since u has the z-component zero, pz = 0, and hence p is +constant in z. Subtracting this constant, we can view p as being zero in the z-component and +in this sense independent of z. So we have proved that u is a solution of the two-dimensional +Stokes equations. +Using polar coordinates, we find +−uy = x sin θ +�f ′ +r − f +r2 +� +, +ux = f ′ sin2 θ + f cos2 θ +r +. +(20) +Impose constraints +f(1) = f ′(1) = 1, +f(b) = f ′(b) = 0. +(21) +The latter two constraints in (21) imply that u = curl ψ = 0 for r = b. The first two +constraints in (21) imply that +u(r = 1) = (1, 0). +Since we have identified four parameters and four constraints, we likely have found the +required solution. But to be sure, we need to solve these equations and see what happens +when b → ∞. +5.2.1 +Algebraic solution of the PDE +Using the boundary conditions (21), we can evaluate the constants A, B, C, and D. We +have +B = +−2(b2 + 1) +2 + 2b2(log b − 1) + 2 log b = +−(1 + b−2) +log b − 1 + b−2(1 + log b) ≈ −1 +log b +and +C = +1 +2 + 2b2(log b − 1) + 2 log b ≈ +1 +2b2 log b, +together with +A = 1 +2B + C +and +D = 1 − 1 +2B − 2C. +Although its derivation is tedious and error-prone, such a result can be checked in various +ways. Thus as b → ∞, +B → 0, b2C → 0 =⇒ A → 0, D → 1. +Therefore ub → (1, 0) as b → ∞. +5.2.2 +Asymptotic behavior +In particular, A, B, and b2C decay like 1/ log b. Using (20), we find +u(r, θ) = (f ′(r), 0) − +� +f ′(r) − r−1f(r) +� +(cos2 θ, cos θ sin θ). +Subtracting the expressions for f ′ and f/r, we find +��f ′(r) − r−1f(r) +�� = +���� +−2A +r2 ++ B + 2Cr2 +���� ≤ +c +log b. +12 + +Examining the expression for f ′, we see that it decays like 1/ log r. Thus we considered the +expression +χb(r) = +� +1 + 3 log r +2 log b +� +f ′(r). +(22) +A plot of χb for b = 10k for k = 2, 3, . . . , 8 is seen in Figure 5. From this figure, we see that +χb ≈ 1 for small r/b. Note that, by definition of χb, +ux ≈ f ′(r) = +� +1 + 3 log r +2 log b +�−1 +χb(r). +(23) +Figure 5: Plot of χb defined in (22) for b = 10k for k = 2, 3, . . . , 8. The horizontal axis is r. +5.3 +Friction boundary conditions +We performed a series of tests solving (4) in the domain (17) with Navier boundary condi- +tions (11) with β = 0, for various r. +Figure 4(b) gives the data for three values of d as a +function of box size r for Navier boundary conditions (11) with β = 0. These data suggest +that ur is converging to (1, 0) as r → ∞ with Navier boundary conditions. +6 +Navier–Stokes: no paradox +According to [6, corollary to Theorem 7A], the nonlinear problem (1) has a solution with +g = 0 and u → u∞ with u∞ a constant, provided that |u∞| is sufficiently small; also see [7, +Theorem XII.5.1]. The realization that adding an advection term to the equations resolves +13 + +0.8 +0.6 +0.4 +0.2 +0 +-0.2 +-0.4 +100 +10 +102 +103 +10° +105 +106 +10° +108the Stokes paradox began with the work of Oseen [19]. See [6] for more historical references. +The results of Finn [6] confirm that, for the Navier–Stokes equations, one can pose boundary +conditions both on the cylinder (or other bluff body) and at infinity. +The constant function g is also a solution of (1) (with constant pressure) for any R > 0. +But the boundary conditions are different in this case. We can sum up the Stokes paradox +by saying that a boundary condition is lost when we set the Reynolds number R to zero. +Thus fluid flow can be described accurately in unbounded domains only by a nonlinear +system. +For the cylinder problem, the diameter L gives us a length scale. Once we pick the flow +u∞ (or g), we have a speed U, and together with the kinematic viscosity ν, this determines +a Reynolds number R > 0 given by (3). The only way R can be zero is to have u∞ = g = 0 +(or infinite viscosity, which does not sound like a fluid). Thus the Stokes equations can be +viewed as an approximation for small Reynolds numbers, and this approximation works well +for bounded domains. But it fails for infinite domains. +The existence of solutions of the Navier–Stokes system for large external flows, or equiv- +alently for large Reynolds numbers, is reviewed by Galdi in [7, section XII.6]. However, the +results there are not definitive; they present a condition that must hold if no such solutions +exist. +7 +Extensions of the Stokes paradox +The Stokes paradox has implications for other flow problems. Here we mention two of them. +7.1 +Flow instability +Determining the form of Reynolds–Orr instability modes for Navier–Stokes flow around a +cylinder requires solution of a generalized eigenproblem of the form [11] +−∆u + ∇p = λ−1BRu in Ω, +∇· u = 0 in Ω, +(24) +with homogeneous boundary conditions on Γ = ∂Ω. Here the multiplication operator BR is +defined by +BR(x) = 1 +2 +� +∇uR(x) + ∇ut +R(x) +� +, +where uR solves (1). Restricted to a bounded domain, this constitutes a symmetric gener- +alized eigenproblem, and thus it has real eigenvalues [25]. +On an unbounded domain, we expect that some rate of decay for B would be required +in order that the eigenproblem is well behaved. Define +V = +� +v ∈ H1 +w(Ω) : v = 0 on Γ +� +, +and we endow V with the norm of H1 +w(Ω). +Lemma 7.1 Suppose that there is a positive constant CB such that +|B(x)| ≤ CB +� +1 + |x|−2 log2 |x| +� +∀x ∈ Ω. +(25) +Then the multiplication operator associated with B is a bounded operator from V to V ′. +14 + +In the statement of the lemma, |B(x)| denotes the Frobenius norm of B(x). To prove +the lemma, recall from [9, page 315] that +∥u∥V ′ = +sup +0̸=v∈V +� +Ω u(x) · v(x) dx +∥v∥H1w(Ω) +. +(26) +But H¨older’s inequality and (25) imply +��� +� +Ω +B(x)u(x) · v(x) dx +��� +2 +≤ +� +Ω +|B(x)| |u(x)|2 dx +� +Ω +|B(x)| |v(x)|2 dx +≤ C2 +B∥u∥2 +H1w(Ω)∥v∥2 +H1w(Ω). +(27) +Thus we conclude that +∥Bu∥V ′ ≤ CB∥u∥H1w(Ω). +This completes the proof of Lemma 7.1. +Consider the operator K defined by Kv = u where u ∈ V solves +−∆u + ∇p = Bv in Ω, +∇· u = 0 in Ω. +(28) +Note that the eigenproblem for K, that is Ku = λu, provides a resolution of (24). The +following is a corollary of Lemma 7.1. +Theorem 7.1 Suppose that (25) holds. Then K is a bounded operator from V to V . +The proof of Theorem 7.1 follows from [9, Theorem 3.4] and Lemma 7.1. +From [7, Remark XII.8.3] we expect that +|∇uR(x)| = O +� +|x|−1 log2 |x| +� +for large |x|. +Thus (25) does not hold for BR, and the associated multiplication operator is not a bounded +operator on H1 +w(Ω). Indeed it was found in [11] that the eigenvalues increase as the compu- +tational domain size is increased. We can summarize these observations as follows. Despite +the fact that the Navier–Stokes equations are well defined on unbounded domains, the +equations for their instabilities are not. We are tempted to call this the instability paradox. +7.2 +Power-law fluids +Tanner [28] has shown that shear thinning power-law fluids do not suffer Stokes’ paradox, +but that shear thickening power-law fluids do. The Stokes power law model is given by [16, +(1.5)] +−ν∇· +� +|Du|r−2Du +� ++ ∇p = f +in Ω, +∇· u = 0 +in Ω, +(29) +where Du = 1 +2 +� +∇u + ∇ut� +. The fluid model is shear thinning if r < 2 and shear thickening +if r > 2. The case r = 2 is the standard Stokes model. +Tanner showed that for flow around a cylinder, the Stokes paradox holds for r > 2, but +not for r < 2. The approach [9] can possibly extend this result to more general domains. +Due to the length of the current paper, we postpone such an investigation to a subsequent +study. +15 + +8 +Numerical implementation +The curved boundary of the cylinder was approximated by polygons Ωh, where the edge +lengths of ∂Ωh are of order h in size. Then conventional finite elements can be employed, +with the various boundary expressions being approximated by appropriate quantities. For +the computations described in section 5.1, we used the Robin-type technique [22] together +with the Scott–Vogelius elements of degree 4. The order of approximation for the numerical +method is h7/2 in the gradient norm. +The remaining results were computed using the lowest-order Taylor–Hood approxi- +mation. +To implement the Navier-slip boundary condition, we used Nitsche’s method +[12, 24, 31] to enforce slip conditions in the limit of small mesh size. The details regarding +numerical implementation of (1) together with boundary conditions (2) and (11), are given +in [12]. The boundary integrals are approximated to order h2, but the order of approxima- +tion for the numerical method is only of order h3/2 in the gradient norm. +9 +Conclusions +We have shown that examining the Stokes paradox from different angles enriches the un- +derstanding of the phenomenon. The approaches dovetail together in the final analysis, but +they allow answers to different questions related to the paradox. Perhaps the most critical +question relates to what goes wrong when we pose the Stokes problem on larger and larger +domains. We explored two different ways to consider this question, via numerical simulation +for general domains and analytical solutions on specific domains. Fortunately, they give the +same advice as to what happens in the limit, and this agrees with the functional analysis +formulation of the problem on an infinite domain. We showed that the Stokes paradox can +arise in other flow problems as well. +10 +Acknowledgments +We thank Vivette Girault for valuable information and advice. +References +[1] F. Alliot and C. Amrouche. Weak solutions for the exterior Stokes problem in weighted +Sobolev spaces. Mathematical Methods in the Applied Sciences, 23(6):575–600, 2000. +[2] Martin Alnæs, Jan Blechta, Johan Hake, August Johansson, Benjamin Kehlet, Anders +Logg, Chris Richardson, Johannes Ring, Marie E. Rognes, and Garth N. Wells. The +FEniCS project version 1.5. Archive of Numerical Software, 3(100), 2015. +[3] Charles J. Amick. On Leray’s problem of steady Navier-Stokes flow past a body in the +plane. Acta Mathematica, 161:71–130, 1988. +[4] an anonymous referee, 2022. +[5] Anis Dhifaoui, Mohamed Meslameni, and Ulrich Razafison. Weighted Hilbert spaces for +the stationary exterior Stokes problem with Navier slip boundary conditions. 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Resolution of d’alembert’s paradox using +slip boundary conditions: The effect of the friction parameter on the drag coefficient. +arXiv e-prints, page arXiv:2204.12240, April 2022. +[11] Ingeborg G. Gjerde and L. Ridgway Scott. Kinetic-energy instability of flows with slip +boundary conditions. Journal of Mathematical Fluid Dynamics, 24(4):1–27, 2022. +[12] Ingeborg G. Gjerde and L. Ridgway Scott. Nitsche’s method for Navier-Stokes equations +with slip boundary conditions. Mathematics of Computation, 91(334):597–622, 2022. +[13] Fr´ed´eric Hecht. New development in freefem++. Journal of numerical mathematics, +20(3-4):251–266, 2012. +[14] Saul Kaplun and P. A. Lagerstrom. Asymptotic expansions of Navier-Stokes solutions +for small Reynolds numbers. Journal of Mathematics and Mechanics, pages 585–593, +1957. +[15] L. D. Landau and E. M. Lifshitz. Fluid Mechanics. Oxford: Pergammon Press, second +edition, 1987. +[16] Lew Lefton and Dongming Wei. 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North-Holland, +third edition, 1984. +[30] Milton Van Dyke. Perturbation methods in fluid mechanics, annotated edition. The +Parabolic Press, Stanford, 1975. +[31] M. Winter, B. Schott, Andre Massing, and W. A. Wall. A Nitsche cut finite element +method for the Oseen problem with general Navier boundary conditions. Computer +Methods in Applied Mechanics and Engineering, 330:220–252, 2018. +18 + diff --git a/4tAyT4oBgHgl3EQfQPYV/content/tmp_files/load_file.txt b/4tAyT4oBgHgl3EQfQPYV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..33c26b203d2cab92c7bdde4c94c7b68b913c1fa9 --- /dev/null +++ b/4tAyT4oBgHgl3EQfQPYV/content/tmp_files/load_file.txt @@ -0,0 +1,578 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf,len=577 +page_content='New Insights on the Stokes Paradox for Flow in Unbounded Domains Ingeborg G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Gjerde and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Ridgway Scott January 3, 2023 Abstract Stokes flow equations, used to model creeping flow, are a commonly used simplifi- cation of the Navier–Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The simplification is valid for flows where the inertial forces are negligible compared to the viscous forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In infinite domains, this simplification leads to a fundamental paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In this work we review the Stokes paradox and present new insights related to recent research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We approach the paradox from three different points of view: modern functional analysis, numerical simulations, and classical analytic techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The first approach yields a novel, rigorous derivation of the paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We also show that relaxing the Stokes no-slip condition (by introducing a Navier’s friction condition) in one case resolves the Stokes paradox but gives rise to d’Alembert’s paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The Stokes paradox has previously been resolved by Oseen, who showed that it is caused by a limited validity of Stokes’ approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We show that the paradox still holds for the Reynolds–Orr equations describing kinetic energy flow instability, meaning that flow instability steadily increases with domain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We refer to this as an instability paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 1 Introduction Fluids like air and water can be modeled to high precision using the Navier–Stokes equa- tions [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' These equations are still intensively studied, and they have some very curious mathematical features, often described as paradoxes, such as d’Alembert’s paradox [10, 26] and Whitehead’s paradox [30, section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Here we discuss one of the most well known, namely the Stokes paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The Stokes paradox also arises for more exotic fluids, such as Fermi electrons [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Consider a domain Ω ⊂ R3 containing an infinitely long cylinder with radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The cylinder has a boundary denoted Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The domain is filled with a fluid moving with velocity u and pressure p past the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Mathematically, this can be described using the Navier– Stokes equations −∆u + ∇p = −R u · ∇u in Ω, ∇· u = 0 in Ω, (1) together with the boundary condition u = g on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (2) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='00039v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='flu-dyn] 30 Dec 2022 In this form, the Navier–Stokes equations are posed with only one parameter, namely the Reynolds number R [15]: R = UL/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (3) Here, U is the representative flow velocity, L the representative length for the domain, and ν is the kinematic fluid viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In this work, we are interested in the case where the Reynolds number will be small, and we have R ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In this case, it seems reasonable to drop the advective term, in which case we have the simpler Stokes equations −∆u + ∇p = 0 in Ω, ∇· u = 0 in Ω, (4) again with a boundary condition of the form (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' If the domain Ω is taken to be infinitely large, however, this simplification leads to a fundamental paradox [27, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The Stokes paradox is that “creeping flow of an incompressible Newtonian fluid around a cylinder in an unbounded fluid has no solution” [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Others have characterized the paradox by saying [23, 1st paragraph] “Stokes (1851) established that there was no solution to the two- dimensional, steady, incompressible, Navier–Stokes equations for asymptotically uni- form flow around a cylinder.” [1, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='15] “no classical solution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' that tends to a nonzero vector at infinity.” These statements require some clarification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Firstly, potential flow around a cylinder solves the Stokes equations with the right boundary conditions at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' However, potential flow (Figure 1a) does not satisfy no-slip boundary conditions on the cylinder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' for this reason it has commonly been discarded as physically incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' To handle the no-slip boundary condition we instead look to the literature on functional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' A precise mathematical theory for the Stokes equations in an unbounded domain was developed in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We can explain the results there informally as follows: That it is a well posed problem to require a reasonable flow profile around a reasonable domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' A solution therefore exists solving Stokes equations in an infinite domain enclosing a cylinder, as long as we set reasonable boundary conditions on the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' But the paradox is that we cannot set a boundary condition at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This point is often misunderstood, saying instead that there is a “need to satisfy two boundary conditions, one on the object and one at infinity” [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This is despite the fact that many papers have dealt with the paradox and its resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The first resolution was by Oseen [19] who realized that it was necessary to keep R > 0 in (1) and provided an approximate (linearized) solution to Navier-Stokes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This observation has been substantially amplified by Finn [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' But many others have dealt with the paradox, for example by improv- ing the Oseen approximation [14, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For a survey and history of their methods, see [30, Chapter V] where the so-called matched asymptotic expansions are traced back to seminal work by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Friedrichs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The Sobolev space in which the solution exists provides important context for the Stokes paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' To be more precise, the solution exists in a special weighted Sobolev space in which we give up control over the function as we get infinitely far from the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For this reason it is not possible to prescribe the flow at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We will show that, if the flow is specified 2 to be a constant on the cylinder, the flow must be that constant everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus, while a non-trivial solution exists, it may not be physically meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Due to the confusion related to the Stokes paradox, we examine it in some detail before proceeding to the main results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We draw together disparate approaches to give the fullest possible understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Although much of what we present at the beginning can be found separately elsewhere, the combination of the different approaches gives a more complete picture than has so far been presented in one place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The article will proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We begin in Section 2 by giving two analytic solutions for Stokes flow around a cylinder and discuss their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Next, we give in Section 3 a derivation of the Stokes paradox, using the mathematical framework from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In Section 4, we examine the impact of a different boundary condition on the cylinder, Navier’s slip (friction) condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We see that it is possible to resolve the Stokes paradox with one value of the friction parameter, although that value seems to be nonphysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We next view the Stokes paradox through other lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In particular, we solve the Stokes problem on a very large (but finite) domain surrounding the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In this case we can pose any boundary conditions we like on the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' So what goes wrong as we let the outer boundary go to infinity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In Section 5 we answer this in two ways, using modern computational methods (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1) and classical analytical solution techniques (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We will see that they give the same answer: the solution goes to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus we have multiple ways of viewing the Stokes paradox, each with its own advantages and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In Section 6 we review the resolution of the Stokes paradox by Oseen [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' According to Oseen, the paradox is caused by the limited validity of Stokes’ approximation, which relies on the Reynolds number being small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' To be more precise, we have two limits going to infinity: the viscosity and the domain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Either limit alone is well behaved, but jointly they are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus one must consider the full Navier-Stokes equations if the domain is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We also discuss briefly some open questions regarding the existence of a solution for the Navier–Stokes equations when the Reynolds number grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In Section 7 we describe extensions of the Stokes paradox concepts to other flow problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In particular, we discuss how the Stokes paradox relates to flow instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The Reynolds- Orr method gives a way to calculate the kinetic energy instability of a perturbed base flow, where the most unstable flow perturbations are calculated by solving a Stokes type eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In [11], it was observed that the instability of a base flow kept increasing as the domain grew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This can be explained as a particular instance of the Stokes paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' To the best of our knowledge, however, it cannot be resolved in any such way as Oseen’s resolution of the Stokes paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 2 Classical solutions Assume for the moment Γ to be an infinitely long cylinder of radius 1 in the z-direction with origin (0, 0) in the x, y-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Consider now the Stokes flow equations (4) with boundary condition (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We now construct two analytic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The strategy is to find a function u that is divergence free so it satisfies the second equation in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' If the Laplacian of u is a gradient of some function, we can solve the first equation in (4) by construction by taking the pressure equal to said function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' To find a function u that is divergence free, we use two different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For the first, we take φ to be a potential function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' we set u = ∇φ = (φx, φy), where φi denotes 3 (a) Potential flow, β = −2ν (b) Zero friction flow, β = 0 Figure 1: Stream function ψ, stream lines and flux u (cones) for two classically known analytic solutions of the Stokes equations for flow past an infinitely long cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Neither solution satisfies the no-slip boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' the derivative of φ in the ith direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Then if we want ∇·u = 0 we want ∇·∇φ = ∆φ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' φ to be harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Many harmonic functions are known in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' we take φ(r, θ) = −(cos θ) � r − 1 r � = −x � 1 − 1 r2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (5) A straightforward calculation shows that φ is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We also have ∆u = (∆φx, ∆φy) = ((∆φ)x, (∆φ)y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus we have a solution to the Stokes equation (4) if the pressure is taken to be a constant (so that ∇p = 0 as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Differentiating r = � x2 + y2 we find rx = x r , ry = y r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (6) Using these, we have φx = 1 − r−2 + 2yr−3ry = 1 + (y2 − x2)r−4, φy = 2yr−3rx = 2xyr−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Then u(x, y) = (φx(x, y), φy(x, y)) = � 1 + y2 − x2 (x2 + y2)2 , −2xy (x2 + y2)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (7) This solution for u is commonly referred to as potential flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The potential flow solution is shown in Figure 1a together with its stream function and streamlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We see that u goes to uniform flow at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Moreover, u · n = 0 on the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' However, the tangential 4 u 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5 4 2 0 2 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5n 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5 0 20 10 0 10 20velocity u · τ ̸= 0, meaning that this solution does not satisfy a no-slip boundary condition on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This led Stokes to reject this solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Let us therefore make another analytic solution, this time trying the second approach to making a divergence-free flux u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' By augmenting our vectors with a z-component we can define u to be the curl of a vector (0, 0, ψ) so that u = (ψy, −ψx, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Dropping the z-coordinate again we have u = (ψy, −ψx) and ∇· u = ψyx − ψxy = 0 as long as ψ is sufficiently smooth1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For example, define ψ = (sin θ) r log r = y log r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (8) Then u(x, y) = �y2 r2 + log(r), −xy r2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (9) By construction u satisfies the second equation in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The first equation in (4) can then be satisfied by choosing p so that ∇p = ∆u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We will show in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2 that such a solution p exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Again, the solution does not satisfy the no-slip boundary condition on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Moreover, the solution diverges to infinity as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In the next section, we see how these analytic solutions are related to the Stokes paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 3 Derivation of Stokes paradox using the variational framework The Stokes paradox occurs for incompressible flow Stokes flow with no-slip boundary condi- tions on the cylinder (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' g = 0 in (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The paradox can be derived in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In this work, we present three approaches: (i) a rigorous derivation using weighted Sobolev spaces, (ii) a formal approach using simulations in domains of increasing size and (iii) a semi-formal approach of deriving analytic solutions in bounded domains and passing to the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1 Sobolev spaces for the Stokes equations If the domain Ω is Lipschitz continuous [8, section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1]), the system is well posed with u ∈ (H1(Ω))d, d ∈ {2, 3} being the dimension of the domain, and p ∈ L2(Ω/R), where L2(Ω)/R = � v : Ω → R : � Ω v2 dx < ∞, � Ω v dx = 0 � is the space of square-integrable functions, together with the norm given by ∥v∥L2(Ω) = �� Ω v2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We also define H1(Ω) = � v ∈ L2(Ω) : ∇v ∈ L2(Ω) � 1for example ψ ∈ C2(Ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' two times continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In this case we can change the order of differentiation without issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 5 and the norm ∥v∥H1(Ω) = �� Ω |∇v(x)|2 + v(x)2 dx, where |v(x)| is the Euclidean norm of ∇v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The notation u ∈ (H1(Ω))d then means that every component of u is in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' To simplify notation, we from now on drop the superscript and simply write u ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In summary, given a bounded domain and reasonable f and g (for example f ∈ L2(Ω) and g ∈ H1(Ω) [8]) there exists a unique solution pair u ∈ H1(Ω) and p ∈ L2(Ω) of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Once we know there exists a unique solution, we can use numerical methods to solve for its approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' If the domain Ω is infinite, the previous result no longer holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Instead, the Stokes equation will be well posed with p ∈ L2(Ω)/R and u ∈ H1 w(Ω) defined by the norm ∥v∥H1w(Ω) = �� Ω |∇v(x)|2 + � 1 + |x| log |x| �−2|v(x)|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (10) Centrally, this is a weaker norm than the one we had for the H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Both spaces require the gradient of the function itself to be square-integrable, but in the H1 w(Ω)-space we only require the function to be square-integrable when multiplied by a weight function � 1+|x| log |x| �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' As this weight function goes to zero as x → ∞, the function does not have to decay at all as we move away from the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' As we will see it may even diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Therefore we have to be very careful about assigning limit values at infinity to functions u ∈ H1 w(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' What can be said, based on [9], is that one cannot specify boundary conditions simulta- neously on the cylinder and at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' That is, having specified conditions on the cylinder, the conditions at infinity have already become specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We will see that the resolution of the Stokes paradox is simple once we know in what function spaces to look for solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We now know that solutions exists for the Stokes problem, but only in a certain weighted Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Due to the weight going to zero as we move away from the cylinder, the solution is allowed to behave more mischievously in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In the time of Stokes (1819–1903), the functional analysis approach to partial differential equations was still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Indeed, it was not until 1991 [9] that the appropriate function spaces were fully clarified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2 Derivation of the Stokes paradox Now that we know there exists a solution, we can straightforwardly formulate the Stokes paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For this, it is useful to choose moving coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Instead of thinking of a fixed cylinder in a moving fluid, let us reverse the point of view by using moving coordinates such that the fluid appears at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' If we think of a moving cylinder in an infinite fluid, we can pose the (Navier–)Stokes equations as in (4) with a boundary function g = (1, 0), assuming the cylinder is moving in the x-direction with unit speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In view of [9], there is a unique solution u ∈ H1 w(Ω), where Ω is the complement of the cylinder (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' {(x, y) ∈ R2 : x2 + y2 > 1}) and fixed in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' But g ∈ H1 w(Ω), and g is a solution of (4), with constant pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' And [9] proves that g is the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus we have proved the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1 Suppose that we move an infinite cylinder in a direction perpendicular to the axis of the cylinder with unit speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' If the entirety of the fluid is governed by the Stokes 6 equations (4) with a no-slip boundary condition on the cylinder, then the entirety of the fluid is forced to move at unit speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In the variational framework, the Stokes paradox is not really a paradox: The Stokes equation is well posed in infinite domains, but the appropriate function space for u is one where we give up control of u as it approaches infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus it is not surprising that the solution is non-physical away from the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The fact that we are not able to specify the limiting value of the solution raises the question of how badly the solution might behave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We investigate this in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1 Limit values of functions in H1 w In the previous section we saw that moving an infinitely long cylinder through an infinite domain Ω with given speed g = (1, 0) caused the entire solution flux to be u = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In fact, due to the weight function, any v ∈ H1 w(Ω) can tend to a nonzero constant at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Worse, it can grow like (log r)α for α < 1/2, as long as its gradient remains square integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In particular, take u(r) = (log r)α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Then for r > 1, |∇u| = ���α∇r r (log r)α−1��� = ���α x r2 (log r)α−1��� = ���α r (log r)α−1���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This expression is square integrable at infinity if � ∞ K r dr r2(log r)2(1−α) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Changing coordinates to s = log r (so that r−1dr = ds), our condition reduces to � ∞ log K ds s2(1−α) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This holds when α < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus the Stokes equation posed in an infinite domain may have a solution that diverges as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' An example of this is the zero friction solution (9), as we now discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 4 Navier’s revenge In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2 we saw how the imposition of a no-slip boundary condition on the cylinder leads to the Stokes paradox in unbounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In this section we will discuss what may happen for different boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We will see that the Stokes paradox does not occur for all boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Instead of the no-slip boundary condition (2) with g = 0, let us consider Navier’s slip condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This boundary condition, sometimes referred to as Navier’s friction condition [18, 12, 5], links the tangential velocity and the shear stress on Γ: β u · τ k = −ν nt(∇u + ∇ut)τ k, k = 1, 2, (11) where τ i are orthogonal tangent vectors and β is the friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This is coupled with the no-penetration condition u·n = 0 on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In our two-dimensional case of flow around a cylinder, there is only one tangent vector τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The other one is perpendicular to the plane 7 of the two-dimensional flow, that is, parallel to the cylinder axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For β > 0, the Navier slip condition works as a friction causing the fluid to slow down as it slips over the cylinder boundary Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The potential function φ and stream function ψ defined in (5) and (8) give rise to two different solutions of the Stokes equations with Navier’s boundary condition (11), each corresponding to a particular choice of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The potential flow solution (7), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' u = (φx, φy) ⇒ u(x, y) = � 1 + y2 − x2 (x2 + y2)2 , −2xy (x2 + y2)2 � , solves the Navier–Stokes equations for all ν, and satisfies the Navier slip condition if β = −2ν [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Moreover, this flow goes to the desired asymptotic limit (zero) at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus (7) resolves Stokes’ paradox for β = −2ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For this particular boundary condition the solution belongs to the standard Sobolev space H1(Ω) and exhibits reasonable physical behavior in the entire domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This raises the question of what happens for other values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Interestingly, the other analytic solution we have, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (9), satisfies the friction boundary condition (11) if β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' To see this, note that τ = (−y, x) and n = (−x, −y) on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Computing (∇u) τ we find (∇u)τ = τ · ∇u = ∂θ � sin2 θ + log(r), − cos θ sin θ � = � 2 sin θ cos θ, − cos2 θ + sin2 θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (12) Therefore (omitting some trigonometric simplifications) nt(∇u)τ|Γ = −(cos θ, sin θ)t� 2 sin θ cos θ, − cos2 θ + sin2 θ � = −2 sin θ cos2 θ + cos2 θ sin θ − sin3 θ = − sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (13) Similarly (∇u)n = n · ∇u = −r∂r � sin2 θ + log(r), − cos θ sin θ � = � − 1, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (14) This says that τ t(∇u)n|Γ = (− sin θ, cos θ) · � − 1, 0 � = sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (15) Note that nt(∇ut)τ = τ t(∇u)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Therefore nt(∇u + ∇ut)τ|Γ = � nt(∇u)τ + τ t(∇u)n � |Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (16) Thus the function in (9) solves the Stokes equations and satisfies the Navier slip condition if β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' But this solution diverges as r → ∞, fast enough that the norm in (10) is not finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus (9) does not resolve the Stokes paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' It is worth noting that the fact that β = 0 does not mean that the drag on the cylinder is zero [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' It is known that the drag IS zero for β = −2ν, and this is the core of d’Alembert’s paradox [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The computations above are sufficiently complex that it is useful to have a way to verify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This can be done by solving the Stokes equations with the Navier slip condition numerically with β = 0 and check that the result approximately agrees with (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For β > 0, the Navier slip condition acts as a friction force, slowing the flow as it slips over the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For β → ∞, the Navier friction boundary condition converges to the Stokes no-slip condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For other values of β, the techniques in section 5 could be used to see if there are plausible solutions of the Stokes paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 8 In conclusion, we see that the Stokes paradox is resolved using the Navier slip boundary condition with one particular value for β, but not for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Navier died in 1836, so he was not available to comment on Stokes’ paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We can only wonder what he might have said.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 5 Stokes flow on bounded domains of increasing size In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1 we saw how the Stokes problem lost the ability to specify the value of the solution on the boundary away from the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' With this in mind, we now restrict our attention to bounded domains, where it is possible to pose boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We explore two approaches, a computational one and an analytical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We will see that as we increase the size of the box, we again encounter the Stokes paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1 Computational approach Recent advances in software [2, 13] have made it easy to solve partial differential equations (PDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Using such software, you can study PDEs without knowing detailed background prerequisites [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We now indicate this approach for the Navier–Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Consider the domain Ωb defined by Ωb = {x : |x| > 1, |xi| < b, i = 1, 2} (17) for b > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Let Γ denote the subset of ∂Ωb defined by Γ = {x : |x| = 1} , that is, Γ represents the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We keep our viewpoint of a cylinder moving through the larger domain where the fluid is at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=', we consider solutions ub of the problem (4) with boundary conditions ub = (1, 0) on Γ, ub = 0 on ∂Ωb\\Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (18) Figure 2 shows the solution for b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Figure 3 shows the horizontal component of the solution for (a) b = 16 and (b) b = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We see that the support of the horizontal component spreads as the box gets bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus we see that the horizontal component of the solutions is not really going to zero at the boundary of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' It remains positive as we go to the edge of the domain both upstream and downstream of the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' To examine how the support of the horizontal component of the solution spreads as b is increased, we considered a functional to examine the size of ub in regions of increasing size d, but fixed independent of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus we defined � Ωb χd(x)2|ub(x)|2 dx � � Ωb χd(x)2 dx (19) where χd(x) is the interpolant on the computational mesh of the cut-off function 1 2 � 1 − tanh � 20 � |x|2 − d2��� 9 Figure 2: Plot of pressure p, flux u (glyphs) and streamlines for the solution the moving cylinder problem (4) in the domain (17) with boundary conditions (18) for b = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Due to no-slip boundary condition u = (0, 0) on the box walls, the fluid is forced to recirculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (a) (b) (c) Figure 3: Plot of the horizontal component of the solution of (4) in the domain (17) with boundary conditions (18) for (a) b = 16, M=64 and (b) b = 32, M=128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' M is the mesh parameter for mshr, with the number of segments for the definition of the circle chosen to be M as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 10 p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='85 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='85 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5 0n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='25n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='25 32 16 0 16 3 uo(x) for b=32 uo(x) for b=16which is very close to 1 inside |x| < d and very close to zero outside of that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' If ub → (1, 0) as b → ∞, then we would expect the expression (19) to increase to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' If on the other hand, if ub → 0 as r → ∞, we would expect the expression (19) to converge to some value less than 1 as b is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Figure 4 gives the data for three values of d as a function of box size b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' It appears (19) indeed increases to 1, which points to ub → (1, 0) for |x| < d as b → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This is in accordance with the Stokes paradox as stated in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' that as b → ∞, we have u = (1, 0) everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' But then the fluid moves like a solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Interestingly, u = (1, 0) satisfies the Navier slip condition with β = 0 since ∇u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus this solution not only satisfies the no-slip boundary condition on the cylinder, but also the Navier friction condition with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The other solution with β = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (9) has different boundary values on the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' It also diverges when r → ∞, unlike the solution u = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (a) (b) Figure 4: Growth of (19) as a function of r (horizontal axis) for three values of d: (top) d = 10, (middle) d = 20, (bottom) d = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (a) Stokes no-slip boundary condition, (b) Navier friction boundary condition, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2 Analytic solutions in bounded domains Let us return from modern numerical software back to the classics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' In this section, we consider analytical solutions in increasingly large circular domains, following [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' These domains are related to the so-called Leray approximate solutions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Following [28, (12)], consider a general biharmonic stream function of the form ψ = f(r) sin θ, f(r) = Ar−1 + Br log r + Cr3 + Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Now let us show that the fact that ψ is biharmonic implies that u = (ψy, −ψx) satisfies the first equation in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For the sake of calculations, let us for the moment augment the domain with a z-component and let ψ = (0, 0, ψ) so that we can define u = curl ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Note that ∇ · ψ = 0 since ψ depends only on x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' By using the vector calculus identity curl (curl v) = ∇(∇ · v) − ∆v, we then see curl ∆u = curl (∆(curl ψ)) = curl curl ∆ψ = ∇ (∇ · ∆ψ) � �� � =0 −∆2ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2 104 102 1030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2 102 103 104Since all four terms in ψ are biharmonic in any open set that excludes the origin, we can then conclude that u satisfies the following: curl ∆u = −∆2ψ = 0 in any open set that excludes the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Invoking Stokes’ theorem [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='9], we conclude that ∆u = ∇p for some scalar function p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus u satisfies (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Since u has the z-component zero, pz = 0, and hence p is constant in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Subtracting this constant, we can view p as being zero in the z-component and in this sense independent of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' So we have proved that u is a solution of the two-dimensional Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Using polar coordinates, we find −uy = x sin θ �f ′ r − f r2 � , ux = f ′ sin2 θ + f cos2 θ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (20) Impose constraints f(1) = f ′(1) = 1, f(b) = f ′(b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (21) The latter two constraints in (21) imply that u = curl ψ = 0 for r = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The first two constraints in (21) imply that u(r = 1) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Since we have identified four parameters and four constraints, we likely have found the required solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' But to be sure, we need to solve these equations and see what happens when b → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1 Algebraic solution of the PDE Using the boundary conditions (21), we can evaluate the constants A, B, C, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We have B = −2(b2 + 1) 2 + 2b2(log b − 1) + 2 log b = −(1 + b−2) log b − 1 + b−2(1 + log b) ≈ −1 log b and C = 1 2 + 2b2(log b − 1) + 2 log b ≈ 1 2b2 log b, together with A = 1 2B + C and D = 1 − 1 2B − 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Although its derivation is tedious and error-prone, such a result can be checked in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus as b → ∞, B → 0, b2C → 0 =⇒ A → 0, D → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Therefore ub → (1, 0) as b → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2 Asymptotic behavior In particular, A, B, and b2C decay like 1/ log b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Using (20), we find u(r, θ) = (f ′(r), 0) − � f ′(r) − r−1f(r) � (cos2 θ, cos θ sin θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Subtracting the expressions for f ′ and f/r, we find ��f ′(r) − r−1f(r) �� = ���� −2A r2 + B + 2Cr2 ���� ≤ c log b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 12 Examining the expression for f ′, we see that it decays like 1/ log r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus we considered the expression χb(r) = � 1 + 3 log r 2 log b � f ′(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (22) A plot of χb for b = 10k for k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' , 8 is seen in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' From this figure, we see that χb ≈ 1 for small r/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Note that, by definition of χb, ux ≈ f ′(r) = � 1 + 3 log r 2 log b �−1 χb(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (23) Figure 5: Plot of χb defined in (22) for b = 10k for k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' , 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The horizontal axis is r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='3 Friction boundary conditions We performed a series of tests solving (4) in the domain (17) with Navier boundary condi- tions (11) with β = 0, for various r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Figure 4(b) gives the data for three values of d as a function of box size r for Navier boundary conditions (11) with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' These data suggest that ur is converging to (1, 0) as r → ∞ with Navier boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 6 Navier–Stokes: no paradox According to [6, corollary to Theorem 7A], the nonlinear problem (1) has a solution with g = 0 and u → u∞ with u∞ a constant, provided that |u∞| is sufficiently small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' also see [7, Theorem XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The realization that adding an advection term to the equations resolves 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='4 100 10 102 103 10° 105 106 10° 108the Stokes paradox began with the work of Oseen [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' See [6] for more historical references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The results of Finn [6] confirm that, for the Navier–Stokes equations, one can pose boundary conditions both on the cylinder (or other bluff body) and at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The constant function g is also a solution of (1) (with constant pressure) for any R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' But the boundary conditions are different in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We can sum up the Stokes paradox by saying that a boundary condition is lost when we set the Reynolds number R to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus fluid flow can be described accurately in unbounded domains only by a nonlinear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For the cylinder problem, the diameter L gives us a length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Once we pick the flow u∞ (or g), we have a speed U, and together with the kinematic viscosity ν, this determines a Reynolds number R > 0 given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The only way R can be zero is to have u∞ = g = 0 (or infinite viscosity, which does not sound like a fluid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus the Stokes equations can be viewed as an approximation for small Reynolds numbers, and this approximation works well for bounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' But it fails for infinite domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The existence of solutions of the Navier–Stokes system for large external flows, or equiv- alently for large Reynolds numbers, is reviewed by Galdi in [7, section XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' However, the results there are not definitive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' they present a condition that must hold if no such solutions exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 7 Extensions of the Stokes paradox The Stokes paradox has implications for other flow problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Here we mention two of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1 Flow instability Determining the form of Reynolds–Orr instability modes for Navier–Stokes flow around a cylinder requires solution of a generalized eigenproblem of the form [11] −∆u + ∇p = λ−1BRu in Ω, ∇· u = 0 in Ω, (24) with homogeneous boundary conditions on Γ = ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Here the multiplication operator BR is defined by BR(x) = 1 2 � ∇uR(x) + ∇ut R(x) � , where uR solves (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Restricted to a bounded domain, this constitutes a symmetric gener- alized eigenproblem, and thus it has real eigenvalues [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' On an unbounded domain, we expect that some rate of decay for B would be required in order that the eigenproblem is well behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Define V = � v ∈ H1 w(Ω) : v = 0 on Γ � , and we endow V with the norm of H1 w(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1 Suppose that there is a positive constant CB such that |B(x)| ≤ CB � 1 + |x|−2 log2 |x| � ∀x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (25) Then the multiplication operator associated with B is a bounded operator from V to V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 14 In the statement of the lemma, |B(x)| denotes the Frobenius norm of B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' To prove the lemma, recall from [9, page 315] that ∥u∥V ′ = sup 0̸=v∈V � Ω u(x) · v(x) dx ∥v∥H1w(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (26) But H¨older’s inequality and (25) imply ��� � Ω B(x)u(x) · v(x) dx ��� 2 ≤ � Ω |B(x)| |u(x)|2 dx � Ω |B(x)| |v(x)|2 dx ≤ C2 B∥u∥2 H1w(Ω)∥v∥2 H1w(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (27) Thus we conclude that ∥Bu∥V ′ ≤ CB∥u∥H1w(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' This completes the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Consider the operator K defined by Kv = u where u ∈ V solves −∆u + ∇p = Bv in Ω, ∇· u = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' (28) Note that the eigenproblem for K, that is Ku = λu, provides a resolution of (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The following is a corollary of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1 Suppose that (25) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Then K is a bounded operator from V to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1 follows from [9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='4] and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' From [7, Remark XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='3] we expect that |∇uR(x)| = O � |x|−1 log2 |x| � for large |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Thus (25) does not hold for BR, and the associated multiplication operator is not a bounded operator on H1 w(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Indeed it was found in [11] that the eigenvalues increase as the compu- tational domain size is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We can summarize these observations as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Despite the fact that the Navier–Stokes equations are well defined on unbounded domains, the equations for their instabilities are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We are tempted to call this the instability paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='2 Power-law fluids Tanner [28] has shown that shear thinning power-law fluids do not suffer Stokes’ paradox, but that shear thickening power-law fluids do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The Stokes power law model is given by [16, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5)] −ν∇· � |Du|r−2Du � + ∇p = f in Ω, ∇· u = 0 in Ω, (29) where Du = 1 2 � ∇u + ∇ut� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The fluid model is shear thinning if r < 2 and shear thickening if r > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The case r = 2 is the standard Stokes model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Tanner showed that for flow around a cylinder, the Stokes paradox holds for r > 2, but not for r < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The approach [9] can possibly extend this result to more general domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Due to the length of the current paper, we postpone such an investigation to a subsequent study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 15 8 Numerical implementation The curved boundary of the cylinder was approximated by polygons Ωh, where the edge lengths of ∂Ωh are of order h in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Then conventional finite elements can be employed, with the various boundary expressions being approximated by appropriate quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' For the computations described in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='1, we used the Robin-type technique [22] together with the Scott–Vogelius elements of degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The order of approximation for the numerical method is h7/2 in the gradient norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The remaining results were computed using the lowest-order Taylor–Hood approxi- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' To implement the Navier-slip boundary condition, we used Nitsche’s method [12, 24, 31] to enforce slip conditions in the limit of small mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The details regarding numerical implementation of (1) together with boundary conditions (2) and (11), are given in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The boundary integrals are approximated to order h2, but the order of approxima- tion for the numerical method is only of order h3/2 in the gradient norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 9 Conclusions We have shown that examining the Stokes paradox from different angles enriches the un- derstanding of the phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The approaches dovetail together in the final analysis, but they allow answers to different questions related to the paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Perhaps the most critical question relates to what goes wrong when we pose the Stokes problem on larger and larger domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We explored two different ways to consider this question, via numerical simulation for general domains and analytical solutions on specific domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Fortunately, they give the same advice as to what happens in the limit, and this agrees with the functional analysis formulation of the problem on an infinite domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' We showed that the Stokes paradox can arise in other flow problems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 10 Acknowledgments We thank Vivette Girault for valuable information and advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Alliot and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Amrouche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Weak solutions for the exterior Stokes problem in weighted Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Mathematical Methods in the Applied Sciences, 23(6):575–600, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' [2] Martin Alnæs, Jan Blechta, Johan Hake, August Johansson, Benjamin Kehlet, Anders Logg, Chris Richardson, Johannes Ring, Marie E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Rognes, and Garth N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The FEniCS project version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Archive of Numerical Software, 3(100), 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' [3] Charles J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Amick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' On Leray’s problem of steady Navier-Stokes flow past a body in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Acta Mathematica, 161:71–130, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' [4] an anonymous referee, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' [5] Anis Dhifaoui, Mohamed Meslameni, and Ulrich Razafison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Weighted Hilbert spaces for the stationary exterior Stokes problem with Navier slip boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Journal of Mathematical Analysis and Applications, 472(2):1846–1871, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 16 [6] Robert Finn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Mathematical questions relating to viscous fluid flow in an exterior do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The Rocky Mountain Journal of Mathematics, 3(1):107–140, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' [7] Giovanni Galdi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' An introduction to the mathematical theory of the Navier-Stokes equa- tions: Steady-state problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Springer Science & Business Media, 2011.' metadata={'source': 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in fluid mechanics, annotated edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' The Parabolic Press, Stanford, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Winter, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Schott, Andre Massing, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' A Nitsche cut finite element method for the Oseen problem with general Navier boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' Computer Methods in Applied Mechanics and Engineering, 330:220–252, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAyT4oBgHgl3EQfQPYV/content/2301.00039v1.pdf'} diff --git a/59E0T4oBgHgl3EQfewCK/vector_store/index.faiss b/59E0T4oBgHgl3EQfewCK/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..173f2be0e8468f698ecdcc3d419959752791e542 --- /dev/null +++ b/59E0T4oBgHgl3EQfewCK/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ee88038f73ff7f9cc908ea4f8fc3e95eb8bdec8ba8fe0bc62a6975861eb6666a +size 7798829 diff --git a/5tFJT4oBgHgl3EQfkyzq/content/tmp_files/2301.11581v1.pdf.txt b/5tFJT4oBgHgl3EQfkyzq/content/tmp_files/2301.11581v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce031d11b9f74c4e302af1d9e6c91594fa92cb2e --- /dev/null +++ b/5tFJT4oBgHgl3EQfkyzq/content/tmp_files/2301.11581v1.pdf.txt @@ -0,0 +1,1187 @@ +A Green(er) World for A.I. +Dan Zhao∗, Nathan C. Frey∗, Joseph McDonald∗, Matthew Hubbell∗, +David Bestor∗, Michael Jones∗, Andrew Prout∗, Vijay Gadepally∗, Siddharth Samsi∗§ +∗ MIT Lincoln Laboratory +©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including +reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or +reuse of any copyrighted component of this work in other works. DOI: 10.1109/IPDPSW55747.2022.00126 +Abstract—As research and practice in artificial intelligence +(A.I.) grow in leaps and bounds, the resources necessary to +sustain and support their operations also grow at an increasing +pace. While innovations and applications from A.I. have brought +significant advances, from applications to vision and natural +language to improvements to fields like medical imaging and +materials engineering, their costs should not be neglected. As we +embrace a world with ever-increasing amounts of data as well as +research & development of A.I. applications, we are sure to face +an ever-mounting energy footprint to sustain these computational +budgets, data storage needs, and more. But, is this sustainable +and, more importantly, what kind of setting is best positioned +to nurture such sustainable A.I. in both research and practice? +In this paper, we outline our outlook for Green A.I.—a more +sustainable, energy-efficient and energy-aware ecosystem for +developing A.I. across the research, computing, and practitioner +communities alike—and the steps required to arrive there. We +present a bird’s eye view of various areas for potential changes +and improvements from the ground floor of AI’s operational +and hardware optimizations for datacenter/HPCs to the current +incentive structures in the world of A.I. research and practice, +and more. We hope these points will spur further discussion, and +action, on some of these issues and their potential solutions. +Index Terms—Green AI, sustainable AI, energy efficiency +I. INTRODUCTION +Issues of environmental sustainability and energy efficiency +have come to center stage as global warming, climate con- +cerns, and their consequences have permeated many aspects +of our economy and society. In finance, sustainable investing +has come to the fore where, in addition to traditional metrics +of assessing risk, themes of environmental, social, and gov- +ernance (ESG) have become important in evaluating financial +and purpose-driven outcomes. Throughout the private sector, +many companies have begun to re-examine and prioritize green +power usage and resource development [1] while governments +have begun to invest heavily in clean energy and climate +resilient infrastructure [2]—the list goes on. +While traditional sources of carbon emissions from agricul- +ture and transportation continue to contribute the lion’s share +of greenhouse gas emissions in the U.S., electricity usage +This material is based upon work supported by the Assistant Secretary of +Defense for Research and Engineering under Air Force Contract No. FA8702- +15-D-0001, and United States Air Force Research Laboratory Cooperative +Agreement Number FA8750-19-2-1000. Any opinions, findings, conclusions +or recommendations expressed in this material are those of the author(s) and +do not necessarily reflect the views of the Assistant Secretary of Defense +for Research and Engineering, or the United States Air Force. The U.S. +Government is authorized to reproduce and distribute reprints for Government +purposes notwithstanding any copyright notation herein. +§Corresponding author. Email : sid@ll.mit.edu +from the operation of supercomputing and data centers are +climbing with historical signs of compute costs and demand +accelerating further in the years ahead [3]. Estimates place +datacenters’ electricity consumption at 1% of global electricity +demand [4] with projections of electricity usage reaching 8%- +21% of global demand by 2030 [5], though extrapolation of +demand trends can be unreliable due to not accounting for new +improvements in energy efficiency [6]. However, even beyond +the energy footprint from electricity consumption, these data- +centers can take up significant amounts of water, either directly +for cooling or indirectly for electricity generation, bearing a +larger than expected environmental footprint—in the U.S., it is +estimated that 20% of datacenter servers’ direct water footprint +is sourced from moderately to highly stressed watersheds +and 50% of servers are at least partially supplied by power +plants in water stressed areas [7]. In addition to the energy +footprint datacenter/HPC operations, embodied carbon costs +[8] such as those associated with manufacturing hardware for +A.I. development and applications also matter, especially as +hardware continues to advance. As such, the environmental +footprint of A.I. may go beyond the costs represented by +carbon emissions of datacenters/HPCs alone. +Fig. 1: Modern AI’s Computational Demands. Note the steep +increase in just the past decade relative to the past 50 years. Source: +OpenAI & The Economist. +As industry adoption and incorporation of algorithms into +products and services become more commonplace, we have +seen significant growth in both the amounts of training data +and the size of the model itself [8] as the main means to +realize performance gains. Simultaneously, fundamental A.I. +research has continued to accomplish increasingly complex +tasks with increasingly complex models and large datasets. +These factors have, in turn, inevitably pushed similar growth +trends in infrastructure investments required to keep pace with +the increased amounts of training, inference, storage, and more +arXiv:2301.11581v1 [cs.AI] 27 Jan 2023 + +Deep and steep +Computing power used in training Al systems +Days spent calculating at one petaflop per second*, log scale +100 +3.4-month +By fundamentals +AlphaGoZerobecomes itsown +doubling +10 +teacher of the game Go +O Language +○ Speech +O Vision +6 +1 +OGames +0 Other +AlexNet, image classification with +0.1 +deep convolutional neural networks +0.01 +0.001 +0.0001 +Two-year doubling +0.00001 +(Moore's Law) +←First era→ +→ Modern era +0.000001 +Perceptron, a simple artificial neural network +0.0000001 +1960 +70 +80 +90 +2000 +10 +20 +Source:OpenAl +*1petaflop=1015calculations(see Fig. 1). Coupled with the anticipated increase in internet +traffic, consumer devices, and demand for the very products +and services some of these algorithms support [5], these +worrying trends in energy demand and its associated energy +footprint are likely to only accelerate. Even so, in the arms- +race for A.I. superiority and operationalization, companies +and institutions involved in A.I. research and its applications +have continually expanded their datacenters and operations. +Google’s datacenter facilities span several countries while +Meta has recently announced the construction of a new A.I. +Research SuperCluster (RSC) claimed to be among the fastest +and largest supercomputing centers upon completion [9]. In +this race to construct an ever-increasing number (and size) of +datacenters, supercomputing clusters, and supporting facilities, +there are few signs that this race will slow down. Instead, com- +panies are accepting this as an inevitability and are looking for +ways to help offset their ever-increasing energy footprint, such +as building their own additional energy production facilities to +fuel their operations [10] [11]. +Energy-efficient data infrastructure and green computing +are hardly new concepts and have seen continued work +and advances. From the development of efficient chips like +Google’s TPUs [12] and other computing efficiency gains +to the application of A.I. algorithms themselves to automate +datacenter operations, there is a long list of existing prac- +tices and current works-in-progress to address the energy- +hungry and data-intensive appetite necessary to sustain these +algorithms. Though these advances in efficiency have kept +pace with the increased computation/energy needs and offset +demand thus far, there may be signs that this is unlikely +to last [13]. There also exists some debate on the true +extent to which issues on A.I.’s sustainability and energy +footprint are accurately described, largely driven by notable +successes in realizing energy and computational efficiency +in model training, datacenter/HPC operation, and hardware. +However, changes in climate resulting in rising temperatures +and more extreme weather patterns are likely to stress cooling +and already strained resources in many areas. While larger, +well-equipped technology companies have the resources and +incentives to act, develop, and adopt efficiently, there are still +clear, unaddressed concerns if all A.I. workflows move to +the same hardware and software stack despite the efficiency +benefits from centralization. As we run up against the limits +of remaining efficiency gains, other ideas and implementations +are needed, either as an anticipatory or preventative measure, +in order to proactively develop strategies that bring the dis- +course to these problems and their potential solutions. +In the following sections, we discuss the prospects of +encouraging energy efficiency across various levels of the +research & development spectra of A.I. and its applications: +(1) the infrastructure and resource utilization level, (2) the +individual user and behavioral level, and (3) the group and +community of A.I. researchers and practitioners at large. These +three aspects cover issues from a micro-to-macro perspective +but also emphasize a key point—no single change on any +one level is likely to be as effective without corresponding +changes on the other levels since these three aspects are part +of a single whole. A concerted, unified effort is required in +order to transition effectively to a greener ecosystem for A.I. +research and practice. To make our analyses more concrete in +our discussions, we leverage data from the MIT SuperCloud +[14], an operational peta-scale HPC system that is actively +used for research, experimentation, and collaborations by the +MIT research community in several disciplines across machine +learning, deep learning, and more. +II. ENERGY & BEHAVIORAL CONSIDERATIONS +In this section, we discuss potential improvements towards +a more energy-aware compute and cluster optimization frame- +work. While we discuss traditional aspects of datacenter/HPC +management in reducing energy expenditure (e.g. hardware, +system-level), we also focus on non-traditional possibilities. +We touch upon issues such as the economic considerations +of energy consumption like the opportunity costs of energy +purchases, the role/effect of user behavior in designing mecha- +nisms to encourage energy-efficient behavior, changes in exist- +ing behaviors (e.g. from either the user side or datacenter/HPC +management side), and combining—but balancing—existing +energy saving mechanisms on the hardware/systems side with +ones accounting for user behavior and incentives. +When it comes to energy efficiency, a simplified optimiza- +tion framework is useful in understanding the objectives, the +available choices/mechanisms are at our disposal to affect +change, their dependence on one another, trade-offs, and more. +This way, we can simplify the overall optimization problem +that operational datacenters/HPCs face: +min +qs,p,c E(qd, qs, p, c, ε) s.t. A(qd, qs, p, c, ε) ≥ α +(1) +where total energy expenditure E(·) and activity level A(·) of +the datacenter/HPC can be affected by various factors: exam- +ples include the “quantity” of compute resources demanded +or currently utilized by users (qd) as well as their usage +behaviors including but not limited to efficient/inefficient +practices, the “quantity” of compute resources supplied or +available to users (qs) and associated resource settings, the +job scheduling system or resource allocation rule in place (p), +control mechanisms (c) such as hardware settings (e.g. power +caps, clock rate settings) or other physical interventions (e.g. +rack placements, cooling setups) and “softer” mechanisms +(e.g. algorithmic, instrumentation) that may be in place, and +ε which accounts for other factors such as temperature (e.g., +ambient, distributions across racks, local climate) and others +(e.g. a datacenter’s fuel mix and energy purchasing patterns, +maintenance schedules, electricity prices and energy mix). +In other words, the goal is to minimize the energy ex- +penditure E(·) of the datacenter subject to a constraint: the +activity or performance level A(·) of the supercluster must be +above some minimum, acceptable threshold α. This constraint +expresses a fundamental trade-off at the heart of energy- +efficiency: reductions in energy consumption or expenditure +need to be weighed against trade-offs in performance (i.e. jobs + +still need to be done at a reasonable pace). If the performance +level constraint α is not satisfied, attempts to reduce energy +expenditure may produce perverse, unintended effects; for +instance, if a change to reduce energy consumption results +in noticeable performance degradation, then users may run +more jobs for longer, producing the opposite effect. Although +one possibility is that higher throughput jobs can reduce total +energy consumption by driving up power consumption but +finishing in shorter periods as a result, we assume here that α +corresponds to a bare minimum performance level—beneath +which even these high throughput jobs contribute little to +the overall energy footprint compared to the other kinds of +workloads/operations present. +Traditionally, resource management in datacenters/HPCs +tends to take an approach closely aligned with the problem +as outlined (Eq. 1), minimizing energy expenditure primarily +through three main ways: adjusting the available “supply” or +amount of resources qs (e.g. number/types of GPUs), adjusting +resource allocation rules and schedulers p, and usage of control +mechanisms c (e.g. hardware settings). These mechanisms can +be quite effective, cheap, and can easily produce intended +results as they do not necessarily require coordination or know- +how from users. While much work has focused on optimiz- +ing energy efficiency through these traditional mechanisms— +affecting available compute resources, resource allocation and +queuing/scheduling rules, or hardware/software and physical +configurations [15] [16]—new sources of efficiency will likely +need to be claimed from ε as we hit diminishing returns and, +eventually, limits from traditional measures. As easy sources +of efficiency are exhausted, these limits will require looking +beyond more traditional levers (i.e., p, qs, c) and towards less- +traditional ones (i.e., qd, ε). +A. Energy, Power, & Opportunity Costs +When considering the energy expenditure or carbon foot- +print of HPCs/datacenters, what quantity should we focus on? +As framed in Eq. 1, the main objective E(·) can represent +any number of quantities correlated with energy expenditure: +kilowatt-hours, power usage effectiveness (PUE), pounds of +CO2 emitted, amount of water used in cooling, etc. Besides +these quantities, E(·) can also account for aspects like the fis- +cal costs of the datacenter’s energy bill or even the opportunity +costs of its choices, arising from the timing, the amounts, or +the fuel composition of its energy demand and usage as well as +how they affect the datacenter’s environmental footprint. The +economic costs of a choice accounts not only for its direct +fiscal or monetary costs, but also its opportunity costs—the +cost of the best alternatives foregone. In this subsection, we +discuss these opportunity costs and strategies to reduce these +costs by changing energy purchasing behaviors like the timing +of energy purchases and other usage patterns. +For instance, consider the usage patterns of the MIT Su- +perCloud system [14] within a given year. Naturally, the +demand and usage of the system’s overall resources will vary +throughout a year, exhibiting regular patterns on different time +scales within the year. Just as demand and load vary, power +2 +4 +6 +8 +10 +12 +Month +200 +250 +300 +350 +400 +450 +Avg. Power (kW) +Power Consumption vs. Sustainable Fuel Generation +5 +6 +7 +8 +% Total from Solar/Wind +Fig. 2: Power Consumption vs. Green Fuel Mix. Average monthly +power consumption of MIT’s E1 hypercluster plotted against monthly +average percentage of supplied total energy derived from solar +and wind (2020-21). There are potential opportunities—high power +consumption when green energy production is low and vice versa +instead of the opposite. +2 +4 +6 +8 +10 +12 +Month +20 +25 +30 +35 +40 +45 +50 +Real Time Avg Price ($/MWh) +Energy Prices vs. Sustainable Fuel Generation +5 +6 +7 +8 +% Total from Solar/Wind +Fig. 3: Energy Prices vs. Green Fuel Mix. Average monthly +energy prices plotted against monthly average percentage of supplied +total energy derived from solar and wind (2020-21). Prices are +monthly locational marginal prices (LMP) from south eastern/central +MA. Note that energy prices tend to be lower when percentage of +sustainable energy is higher. +consumption will also vary—more users and jobs generally +translate into more computation and increased cooling costs, +increasing power draw from existing resources. Beyond the +dollars-and-cents of the HPC’s electricity bills, the make-up +or composition of the energy supplied by the power company +via the local grid can also influence the sustainability of a +datacenter/HPC’s operations albeit in a less direct way. The +different sources from which power is generated (i.e. the +fuel mix), supplied to, and consumed by the HPC carry an +implicit environmental opportunity cost: the usage or purchase +of power with a less sustainable fuel mix at a period in +time forgoes usage of power generated with a greener fuel +mix in that same time period. This, in turn, represents the +foregone opportunity to offset some portion of existing energy +expenditure while imposing an environment cost in the form +of greater energy inefficiency as an externality. One way to +then improve energy efficiency is to shift energy expenditure +more towards power sourced from higher ratios of sustainable +fuel mixes (i.e. generated with more sustainable sources like +solar and wind). +Figure 2 suggests there may be an opportunity to change +the datacenter/HPC’s purchasing behavior for this strategy to +be viable. Over the course of the year, we see that the total +share of fuel/energy produced from solar and wind is inversely + +related to the average amount of power used per month. The +MIT Supercloud energy consumption has been relatively high +when the share of renewable energy is low around June to +August—similarly, energy consumption/expenditure is lower +when the share of renewable energy in the fuel mix of the +power supplied is higher. One strategy to take advantage of this +mis-match between power consumption and fuel mix, increase +energy efficiency, and reduce the environmental opportunity +cost is to purchase more power during times when sustainable +energy takes up a larger share of the fuel mix (e.g. March +to May) and either: (1) capitalize during that time period by +encouraging more cluster utilization during those months or +(2) store that energy to help offset energy consumption during +times where the fuel mix is less sustainably sourced. +Figure 3 suggests this strategy also carries financial ben- +efits. During springtime, from February to May, when the +sustainable energy share of fuel mix tends to be high (> 8%), +general energy prices tend to be extremely low ($20-$25 per +megawatt-hour) and are some of the lowest prices of the +year. However, it is important to note that renewable energies +like solar and wind may not always see stable generation; +moreover, there are additional fixed costs incurred from setting +up the relevant infrastructure that may be required in order to +pursue strategies like the ones described above. We explore +and discuss the application of A.I. to help stabilize sustainable +energy generation as well as infrastructure investments as they +relate to efficiency in the sections below. +B. Temperature-aware & Weatherized Compute Optimization +While changes in the regular, shorter-term behavior of +datacenters/HPCs can be helpful, like those described above, +longer-term structural changes and preparations are essential. +As changes in climate produce increasingly extreme weather +events and rising temperatures [17], traditional mechanisms +alone may be insufficient to brace for what is to come. +In light of these upcoming challenges, energy-aware cluster +optimization must find ways to explicitly account for factors +in ε that, though difficult to anticipate, carry significant +consequences to datacenter/HPC health and efficiency such +as weather and climate. How would existing concepts and +practices of cluster management and energy efficiency change +with more extreme climate and more frequent weather events? +What would weatherized compute optimization look like? +In Fig. 4, we see the monthly average temperature and +its trend along with those of power consumption for the +MIT Supercloud system. Throughout the year, there is a +monotonic, one-to-one relationship between average monthly +power consumption and average monthly (local) temperature. +As temperatures become warmer heading into the spring and +summer months, it takes more power to cool the facilities +and maintain a sufficiently low temperature for normal oper- +ations, resulting in increased power consumption. If average +temperatures continue to climb even in the colder months as +a consequence of climate change, cooling is likely to become +more difficult and costly as previously efficient mechanisms +for cooling facilities may suffer previously unseen stress. +Fig. 4: Power Consumption vs. Green Fuel Mix. Average monthly +power consumption of MIT Supercloud plotted against monthly aver- +age temperature (in Fahrenheit). Note the near one-to-one relationship +between temperature and power consumption. +As such, investments into infrastructure weatherization is +critical. As changes in climate induce more extreme weather +events and temperature ranges with increasing regularity, ex- +isting methods to realize energy efficiency may no longer be +as effective under more frequent or extreme weather/climate +conditions especially if mechanisms only function effectively +within a small band of temperature/climate conditions. Since +historical data points of extreme weather can be rare (for now), +a useful exercise can be a regularly conducted stress-test akin +to the Dodd-Frank stress tests [18] enacted after the 2008 +financial crisis; these stress tests are conducted annually and +provide simulated stress scenarios that test the resiliency of +financial institutions in both its traditional functions/operations +as well as with less traditional risks (e.g. geopolitical, climate, +infrastructure), helping identify areas in need of remediation. +Similar stress scenarios and risk identification, conducted +and evaluated regularly, for not just regular datacenter/HPC +operations but also for climate and weather resiliency can +help anticipate what energy efficiency (and inefficiency) looks +like when considering future changes in weather and climate. +For institutions with more than one HPC/datacenter, these +exercises can provide opportunities to plan and coordinate +across geo-scattered HPCs/datacenters to improve their col- +lective resilience or develop re-routing backups in extreme +weather conditions. Most importantly, these exercises can help +anticipate and identify critical areas of infrastructure which +require both a significant time and financial investment that +may not come up otherwise. +C. Incentives, Behavior, & Mechanism Design +Hardware and system-level mechanisms can carry much of +the weight in producing energy savings under-the-hood and +abstracting away difficulties without taking away from user +experience. If these interventions run into diminishing returns, +then discovering remaining gains in efficiency will require +work not only from the “supply” side of computing but also +on the “demand” side, qd—the user. Compared to the macro- +level approach dealing with cluster/datacenter-wide hardware +and system-level interventions, this micro-level approach can + +PowerConsumptionvs.MonthlyTemperature +450 +70 +400 +Avg.MonthlyTemperature (F) +Power (kW) +350 +60 +300 +50 +250 +40 +200 +2 +4 +6 +8 +10 +12 +Monthprovide additional flexibility but will require careful planning +around mechanism design, user behavior, and user incentives. +From this perspective, the optimization problem faced by the +datacenter changes from Eq. 1 to +min +i +ei(qd(i), qs, p, c, ε) s.t. ai(qd(i), qs, p, c, ε) ≥ αi ∀i +where +� +i +ei = E, +� +i +ai = A +(2) +for each individual or representative user (or workload) i. +Whereas before the datacenter/HPC in Eq. 1 had control +mainly through qs, p, and c, now the main mechanism is +through a specific user/profile/representative workload i. This +ultimately translates into the datacenter attempting to induce +changes in the quantity of resources demanded qd, as reflected +by qd(i). Instead of total across-the-board quantities like total +energy and total activity/performance, E(·) and A(·), we +now focus on individual (or representative) users, profiles, or +representative workloads and their energy usage and activity +profiles, as denoted by ei(·), ai(·), and αi. Naturally, by tailor- +ing energy minimization efforts to representative user profiles +and workloads, these mechanisms can reduce overall energy +expenditure selectively in ways that systematic hardware inter- +ventions cannot. These micro-level approaches aim to induce +behavioral changes in users through affecting incentives with +the support of predictive analytics and instrumentation. +One example is the design of queues for finer user and +workload segmentation; these queues can improve job schedul- +ing and execution using user-provided information (and other +information) like the user’s stated preferences on energy +efficiency, job urgency/patience, expected time completion, +type of workload, etc. Policies can then be tailored more +specifically with only the resources necessary, allowing for +more efficient design elements by reducing idle time, over- +allocation, and over-utilization of resources. However, if queue +selection and user intent conflict in situations where the user +has an incentive towards a specific resource configuration +different from the assigned one, this mechanism runs the risk +of adverse selection—users mis-characterize their preferences +and select themselves into queues where resources are fastest, +most plentiful, or the most available, leaving select queues +clogged and overtaxed and others largely, if not entirely, idle. +In the example above, too many self-characterizing choices +are made available for users to potentially mis-represent their +preferences and extract private benefits while imposing a social +cost on the whole system. One alternative to balance these two +factors of too much choice and too little control is to maintain +a two-part mechanism: a fixed component that guarantees a +specified minimum amount of energy efficiency and a variable +component that allows for user choice to further scale energy +efficient behavior, but only in certain respects. For instance, +it has been shown that optimal GPU power-caps provide an +effective way to control energy consumption with minimal +impact on training speed [15] and user experience. With these +optimal power caps as the fixed base component, the variable +component can be offered as a choice: if an user accepts +increasingly stringent power caps on his/her allocated GPUs +(or other restrictions), the user can then, in exchange, choose +to have more GPUs allocated to his/her tasks. These types of +choice mechanisms require a cost-benefit analyses to balance +individual net benefits/costs with system-level benefits/costs +but can help induce energy-efficient changes in user behavior +and computing demand. +Designing mechanisms can be difficult but predictive mod- +els and analytical tools can help in understanding and evaluat- +ing both utilization patterns as well as opportunities to affect +them in an energy-efficient way. Models that help forecast +and relate energy prices, fuel mix, as well as energy expen- +diture to one another can provide significant support in the +decision-making process for optimizing energy purchases and +consumption. Similarly, models leveraging data on compute +demand and usage (e.g. holidays, research deadlines) can help +with scheduling, maintenance, etc. Though these mechanisms +are not without their drawbacks, predictive analytics and in- +strumentation can help mitigate these shortfalls by anticipating +and analyzing behavior via data and inference. +III. CLIMATE-AWARE RESEARCH ECOSYSTEMS +A significant part of the A.I. research ecosystem is driven +and structured by incentives to publish in notable, high- +visibility conferences and journals. These venues serve as +important forums for the A.I. community—researchers, prac- +titioners, and the state of research as a whole—to disseminate +new and important findings, promote brands, seek/hire talent, +highlight significant contributions and problems, exchange +information, foster innovation and collaborative relationships, +and more. These contributions notwithstanding, the way the +research ecosystem is currently structured can create incen- +tives worth reconsidering when transitioning towards a more +sustainable research environment. +As both fundamental research and applications in A.I. to +various fields continue to grow, high-visibility venues will +likely receive more focus and submissions as researchers and +practitioners strive to publish in the “best” possible venue. +Many metrics of success in fundamental and applied research +are also heavily influenced, if not defined, by publishing +in these venues—preferring or requiring that researchers, +practitioners, and even job candidates to have publications +at notable venues—which continues to serve as a common +incentive and evaluative criterion. With such a significant focus +on publication in key conferences, how do these incentives +drive the pattern of research activity and what environmental +consequences do they carry, if any? Previous works have +studied the carbon footprint generated by participants traveling +to conferences [19] [20] but less attention has focused on the +effect of the distribution of deadlines themselves. +Conferences deadlines are typically scattered throughout the +year with each conference serving a specific domain or as +a general purpose venue (e.g. see Table I). Specific dates +are publicized several months ahead to give enough time for +preparation and planning. The distribution of these deadlines +may induce certain patterns in aggregate research activity, + +compute demand, and therefore energy utilization, the last +of which we use as a proxy for activity/demand. As an +exploratory analysis, we compare the number of conference +deadlines per month from January 2020 to end of year 2021 +with trends in monthly energy usage in the MIT Supercloud +system (Figure 5). To help account for the confounding +effects of seasonality, temperature, and other factors on energy +utilization, we include data across two years (2021 & 2022). +Given the way deadlines are structured, we might expect +a lagging relationship where activity or compute demand, +and hence energy utilization, might pick up in anticipation of +upcoming deadlines—the larger the number or concentration +of upcoming deadlines, the larger the increase in compute +demand. As deadlines approach, users are accelerating their +workloads, finishing or repeating experiments, and preparing +for conference submission. In Figure 5, we see some pick- +up in energy usage leading up to the months with a high +concentration of deadlines (i.e. July 2020)—such as the uptick +starting around March/April 2020 and leading up to July +2020—but this may also be due to higher temperatures and +cooling costs as noted earlier. However, there is a sharper +pickup in energy usage starting around Jan/Feb 2021 in +anticipation of a notable concentration of deadlines in the +subsequent months. This sharp increase in energy usage is +significantly higher than in the same period of the previous +year despite no significant differences in average temperature +or other known factors in those time periods between the two +years—the only difference being the concentration/number of +deadlines. Overall, we also see that many deadlines tend to +concentrate in the spring/summer across both years when the +combination of higher temperatures and increased compute +demand can exacerbate existing energy trends, resulting in +significantly higher energy usage that taxes the cluster. In the +same period (i.e. the summer months), the fuel mix of the +supplied power also has the lowest ratio of sustainable energy +of the year, as seen earlier (Fig. 2), which further contributes +to an enlarged environmental footprint. +A natural question that may arise is: can we structure +deadlines to spread out energy utilization and compute demand +to benefit energy efficiency? If the same amount of compute +is to be spent throughout an representative year of research +activity regardless, then several options may help distribute +that amount in a more sustainable fashion: (1) spread deadlines +more uniformly throughout the year, (2) concentrate deadlines +in the winter/spring months when preceding months are colder +or see more sustainable fuel generation, or (3) abolish fixed +deadlines in favor of rolling submissions. Some venues (e.g. +Transactions on Machine Learning Research) have already +shifted to rolling submissions albeit for different reasons. +We note that our preliminary analysis is intentionally limited +in scope as we focus exclusively on the MIT Supercloud +system. Additionally, it neither accounts for other confounding +factors explicitly nor does it show a definitive connection be- +tween conference timings and usage/energy intensity. Rather, +it is meant to bring attention to how structural incentives +in the current A.I. research ecosystem and community may +not align optimally with desirable aspects of sustainability— +with one example being conference deadlines. More work and +data are required to tease out the full picture of the degree +to which aggregate research activity and its energy footprint +are affected by conference timings. We hope that future +work will undertake a finer analysis, accounting for details +such as workload type, type of research activity represented, +breakdown of activity and energy use by domain (e.g. NLP), +etc. beyond just data from this cluster. This requires more data, +better data, data access, as well as willingness to share these +data, which may not currently exist in sufficient amounts, a +matter we discuss further below. +TABLE I: List of notable conferences. The following con- +ferences are considered for analysis (not exhaustive). +Area/Discipline +Conferences +NLP/Speech +EACL, InterSpeech, EMNLP, AKBC, ICASSP +ISMIR, AACL-IJCNLP, COLING, CoNNL, +WMT, EACL +Computer Vision +ICME, ICIP, SIGGRAPH, MIDL, ICCV, +FG, ICMI, BMVC, WACV +Robotics +IROS, RRS, CoRL, ICRA +General ML +COLT, ICCC, ICPR, AAMAS, AISTATS, CHIL +EMCL-PKDD, NeurIPS, ACML, AAAI, ICLR +Data Mining +SDM, KDD, SIGIR, RecSys, CIKM, ICDM +WSDM, WWW +Fig. 5: Energy Usage vs. Number of Conference Deadlines +Average monthly power consumption of MIT’s E1 cluster plotted +against number of monthly conference deadlines (Table I) +IV. CLIMATE-AWARE RESEARCH PRIORITIES +A discussion on the sustainability of the current A.I. re- +search ecosystem and its incentives would be incomplete +without discussing the thematic lines of work, both old and +new, such an ecosystem should prioritize in order to improve +its sustainability and keep its environmental footprint small. +A. Novelty, Redundancies, & Efficiency +Given the complexity and variety of research and appli- +cations in A.I., there are likely significant redundancies in +A.I. workflows. Many experiments usually begin with training + +Energy Usage vs.Conference Deadlines +700 +EnergyUsage +6 +600 +5 +Conference Deadlines +(Avg. Power kW) +500 +4 +400 +3 +300 +2 +200 +1 +2020 +6-2020 +9-2020 +-2020 +-2020 +2021 +8-2021 +10-2021 +6known and proven models up to some pre-specified level +of performance, depending on the research direction, before +building atop these results. Doing so may require some hyper- +parameter search, if not full-blown optimization, resulting in +multiple training runs and inevitably redundant runs, wasted +compute, and additional energy costs. Some redundancies can +play a helpful role by training students and researchers when +they start working on A.I. research where experience obtained +from reproducing results can help shape best practices down +the road. However, problems with reproducability of research +only compound these redundancies as (multiple) attempts at +replication also waste resources and energy when researchers +and practitioners attempt to build off existing work or put +previous work into practice. These difficulties in replicating +published results are wide-spread and well-documented [21], +resulting from inconsistent reporting of sensitivity to hyper- +parameters and training settings (or complete lack thereof), +poor communication, missed opportunities from reviewers, +mis-representation, or some combination of the above. +In the ever-changing landscape of new research and model +frameworks, problems with redundancy and reproducibility +can carry additional implications for energy efficiency. If +incentives to develop better performing models overshadow +those for reproducibility and transparency, research efforts +devoted to producing newer, better models will outpace efforts +for clearer benchmarking and reporting, leaving transparency +and resource efficiency efforts forever playing catch-up. For +instance, when GPT-3 debuted, despite its impressive perfor- +mance on generative language tasks, its training (not including +experimentation during its development) was prohibitively +costly and estimated at around $5 million using a specially +designed supercomputer by Microsoft [22], making it very +difficult for researchers to train and test on their own— +only after its introduction, extensive usage, and popularization +did work focus addressing its efficiency and other issues +(e.g. safety, A.I. alignment, etc.). Over-parameterization and +big data may offer easy performance improvements, but an +emphasis on jointly co-optimizing efficiency and performance +in research may help avoid this efficiency-in-hindsight ap- +proach and front-loading significant energy costs in model +development. Some progress has been made in addressing +these problems as Google, Meta, and other large players have +highlighted best practices and standards that have helped to +significantly reduce their own carbon footprints [23] [8] for +state-of-the-art NLP models, such as efficient model selections +and hardware/system choices. Despite this, however, the fun- +damental problem of information reporting and data availabil- +ity still remains. To remedy this, there needs to be an active, +systematic, and consistent approach towards collecting and +reporting data/information (on energy usage, training settings, +etc.) that incentivizes voluntary contribution and surveys a +sufficiently broad swath of sources to be representative of the +diversity of workloads in research and practice. +B. Measurement, Reporting, & Transparency +Various works have produced estimates in attempts to +quantify the carbon or energy footprint of deep learning +model training with estimates ranging from as high as 5x the +average lifetime emissions of a car [24] to as low as 10−5 +times that amount [23] for state-of-the-art transformers. These +estimates are inherently variable and difficult—not only due +to differences in aspects like hardware (e.g. GPU vs. TPU)— +in both the approach taken to quantify these costs and their +resulting accuracy. These difficulties in accurate estimation +highlight the importance of regularly detailing energy usage +and other information in research alongside typical items like +performance results and ablation tests. Moreover, while many +estimates have focused on training costs, even less clear are +the costs arising through a model’s entire life-cycle, which are +particularly important in industry and applied settings. Even +so, there exist even less data on the costs of inference. +The discrepancies in, and even availability of, these esti- +mates can be due to several reasons. The first is resource +asymmetry—not only do different companies, groups, and +individuals have different amounts of computational resources, +they also have different computational setups so certain met- +rics and calculations may naturally vary depending on the +underlying technological stack. This differentiation similarly +applies in academic disciplines where a base model (e.g. +graph neural networks) may branch out into highly special- +ized, differentiated variants depending on the field or task +(e.g. social networks vs. molecular predictions), resulting in +significantly different training procedures, learning dynam- +ics, energy footprints, and more. Different needs, resources, +and constraints largely determine variations across research +and development workflows; as such, when a company or +institution reports realized gains in efficiency or savings, +these gains may only be realizable on their systems, with +their resources/hardware/configuration, or limited to a specific +class of models that are reported by, or essential to, said +organization. Though a seemingly simple solution would be +to move over to services provided by organizations with the +hardware and technical capabilities to realize such efficiencies, +there are ethical concerns and market concentration issues that +require addressing. Even with similar tasks across companies +and industries, different domains are also characterized by +other considerations and constraints such as the lack of tech- +nical expertise, specific resource and regulatory constraints, +and other requirements like model privacy or interpretability +that may outweigh model performance and efficiency. At +its worst, resource asymmetries can hamper reproducibility +and verification efforts: if state-of-the-art models developed +by large, well-equipped research groups are too costly and +resource-intensive to train for others, how can their results +and estimates be reproduced or verified? +Along with the resource asymmetry, information asymmetry +can discourage and dis-incentivize researchers and practition- +ers from reporting necessary or relevant information. Some +examples of these asymmetries, besides ones mentioned earlier + +like inconsistent reporting of training settings as well as poor +communication and presentation of research results, can arise +in part from incentives to preserve competitive advantages +and other sensitive information. Incentives to protect and +preserve a competitive edge from peers and competitors can +discourage full, transparent reporting of information especially +if these models and research tie into a company’s products +and services. Even when reporting, these incentives may +limit the amount of information made available to the wider +research community, leading to confusion around estimates +and methodologies. Incentives to keep information, and its +benefits, private for competitive advantage can lead to con- +tinued information asymmetries in a self-reinforcing cycle. +Voluntary reporting may then be dominated by larger, better- +equipped groups with the resources and technical ability to +optimize their operations which, though well-intentioned, will +likely not reflect the true extent of the overall, or even the +average, environmental footprint of A.I. and its applications. +Moreover, despite the focus on the footprint and costs of +training, data and estimates on inference are even scarcer +despite its significance—the few estimates, where available, +put inference at 90% of production ML infrastructure costs +[25] and 80%-90% of energy costs [26]. While training enjoys +scaling benefits that saturate GPUs, the different performance +requirements of inference can result in poor GPU utilization +since inference queries are unable to realize the parallelism +that offline mini-batch training enjoys [27]. Low resource- +efficiency and utilization is quite common: AWS reports p3 +GPU instances at only 10%-30% utilization [25] and even +Google’s TPUs exhibit a utilization of 28% on average [28]. +The issues outlined above all point to a common set of +problems that require (1) a better, more representative idea of +the kind of A.I. models, and the underlying resources, used +across disciplines, domains, and communities, (2) a common +set of meaningful metrics, and (3) incentives through both +existing avenues (e.g. conferences, papers) and new ones such +as forums, competitions, leaderboards, or open challenges to +encourage reporting of energy/utilization data and develop- +ment of more energy-efficient models rather than just better +performing ones. To accurately quantify the environmental +footprint, it is essential to capture costs with metrics that +realistically reflect and represent the workloads undertaken in +A.I. research and practice—as well as the burdens and en- +ergy footprint associated with state-of-the-art models on more +representative computational setups rather than in the most +efficient, advanced settings. To incentivize consistent reporting +and sharing of data, the research community needs forums +that prioritize energy-efficient models and methodologies. For +instance, a Green A.I. challenge (in development) that aims to +cast the problem explicitly by challenging participants to max- +imize performance given explicit training and energy budgets. +Lastly, facilities should also provide the central infrastructure, +user interfaces, and analytical tools/instrumentation/logging +to further encourage easy reporting and sharing of data, +especially since not all users are equipped with the expertise +to manually report relevant data and information. +C. A.I. for Energy Savings, Generation, & Discovery +Despite its potential environmental footprint, some of the +most impressive applications of A.I. algorithms have included +ones that help generate energy savings themselves. One exam- +ple has been Google and DeepMind’s use of neural networks to +monitor and optimize their datacenters, reducing the amount of +energy spent for cooling by 40% and PUE by 15% in live tests +[29]. Similar examples abound, but beyond energy savings, +continued and improved sustainability will also require work +from the other side of the equation: energy generation. +The study and application of A.I. to energy discovery and +generation should be strongly incentivized given its immediate +benefits. Current examples include the application of algo- +rithms to stabilize and boost sustainable energy generation: +wind farms provide inexpensive, carbon-free energy but can +be unpredictable, making planning and energy delivery/storage +difficult. In response, DeepMind has developed neural net- +works trained on weather forecasts and historical turbine +data to forecast energy output 36 hours ahead, making early +recommendations on optimal hourly delivery commitments +to the grid possible [30]. Beyond existing energy sources, +A.I. research can help push forward new sustainable energy +sources. Recent work has shown how deep reinforcement +learning can help control nuclear fusion [31] by learning to +control and change the shape of plasma via manipulation of +its magnetic field. Scientific collaborations, especially as they +relate to development of new energy sources or improvements +in existing energy generation, should receive equal priority +and recognition as state-of-the-art performance improvements +in areas like vision and NLP. To do so, partnerships with +scientific and energy researchers should be encouraged and +made more accessible to A.I. researchers and practitioners. +Similarly, benchmark energy datasets should be constructed +and made easily accessible just like standard data benchmarks +in NLP and vision—moreover, these energy datasets should +receive continuous updates and testing due to the inherently +variable behavior of wind, weather, etc. +V. CONCLUSION +There are many dimensions of this multi-faceted problem +that are not addressed in this paper due to space limitations +but are important for consideration nonetheless such as the +equity and accessibility aspects of energy-efficient computing. +Though daunting, we hope our discussions of these problems +and their potential solutions will provide a framework that +spurs further discussion, and most importantly action, on these +various issues. +ACKNOWLEDGMENT +The authors acknowledge the MIT SuperCloud [14] and +Lincoln Laboratory Supercomputing Center for providing HPC +and consultation resources that have contributed to the research +results reported within this paper. The authors acknowledge +the MIT SuperCloud team: William Arcand, William Berg- +eron, Chansup Byun, Michael Houle, Jeremy Kepner, Anna +Klein, Peter Michaleas, Lauren Milechin, Julie Mullen, Albert + +Reuther, Antonio Rosa, and Charles Yee. The authors also +wish to acknowledge the following individuals for their con- +tributions and support: Bob Bond, Allan Vanterpool, Tucker +Hamilton, Jeff Gottschalk, Tim Kraska, Mike Kanaan, Charles +Leiserson, Dave Martinez, John Radovan, Steve Rejto, Daniela +Rus, Marc Zissman. +REFERENCES +[1] U.S. Environmental Protection Agency, +“Green +power +partnership +national top 100,” 2022. [Online]. Available: https://www.epa.gov/ +greenpower/green-power-partnership-national-top-100 +[2] The White House, +“Fact +sheet: +The +bipartisan +infrastructure +deal +boosts +clean +energy +jobs, +strengthens +resilience, +and +advances +environmental +justice,” +2021. +[Online]. +Available: +https://www.whitehouse.gov/briefing-room/statements-releases/2021/ +11/08/fact-sheet-the-bipartisan-infrastructure-deal-boosts-clean-ener\ +gy-jobs-strengthens-resilience-and-advances-environmental-justice/ +[3] J. Sevilla, L. Heim, A. 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Buchli et al., “Magnetic control of tokamak +plasmas through deep reinforcement learning,” Nature, vol. 602, no. +7897, pp. 414–419, 2022. + diff --git a/5tFJT4oBgHgl3EQfkyzq/content/tmp_files/load_file.txt b/5tFJT4oBgHgl3EQfkyzq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e6370512a1dd1cb281e26a83bdc05eeb61e9a1a9 --- /dev/null +++ b/5tFJT4oBgHgl3EQfkyzq/content/tmp_files/load_file.txt @@ -0,0 +1,592 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf,len=591 +page_content='A Green(er) World for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Dan Zhao∗, Nathan C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Frey∗, Joseph McDonald∗, Matthew Hubbell∗, David Bestor∗, Michael Jones∗, Andrew Prout∗, Vijay Gadepally∗, Siddharth Samsi∗§ ∗ MIT Lincoln Laboratory ©2022 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='1109/IPDPSW55747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='00126 Abstract—As research and practice in artificial intelligence (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=') grow in leaps and bounds, the resources necessary to sustain and support their operations also grow at an increasing pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' While innovations and applications from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' have brought significant advances, from applications to vision and natural language to improvements to fields like medical imaging and materials engineering, their costs should not be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As we embrace a world with ever-increasing amounts of data as well as research & development of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' applications, we are sure to face an ever-mounting energy footprint to sustain these computational budgets, data storage needs, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' But, is this sustainable and, more importantly, what kind of setting is best positioned to nurture such sustainable A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' in both research and practice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In this paper, we outline our outlook for Green A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='—a more sustainable, energy-efficient and energy-aware ecosystem for developing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' across the research, computing, and practitioner communities alike—and the steps required to arrive there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' We present a bird’s eye view of various areas for potential changes and improvements from the ground floor of AI’s operational and hardware optimizations for datacenter/HPCs to the current incentive structures in the world of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' research and practice, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' We hope these points will spur further discussion, and action, on some of these issues and their potential solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Index Terms—Green AI, sustainable AI, energy efficiency I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' INTRODUCTION Issues of environmental sustainability and energy efficiency have come to center stage as global warming, climate con- cerns, and their consequences have permeated many aspects of our economy and society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In finance, sustainable investing has come to the fore where, in addition to traditional metrics of assessing risk, themes of environmental, social, and gov- ernance (ESG) have become important in evaluating financial and purpose-driven outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Throughout the private sector, many companies have begun to re-examine and prioritize green power usage and resource development [1] while governments have begun to invest heavily in clean energy and climate resilient infrastructure [2]—the list goes on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' While traditional sources of carbon emissions from agricul- ture and transportation continue to contribute the lion’s share of greenhouse gas emissions in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=', electricity usage This material is based upon work supported by the Assistant Secretary of Defense for Research and Engineering under Air Force Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' FA8702- 15-D-0001, and United States Air Force Research Laboratory Cooperative Agreement Number FA8750-19-2-1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Assistant Secretary of Defense for Research and Engineering, or the United States Air Force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' §Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Email : sid@ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='edu from the operation of supercomputing and data centers are climbing with historical signs of compute costs and demand accelerating further in the years ahead [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Estimates place datacenters’ electricity consumption at 1% of global electricity demand [4] with projections of electricity usage reaching 8%- 21% of global demand by 2030 [5], though extrapolation of demand trends can be unreliable due to not accounting for new improvements in energy efficiency [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' However, even beyond the energy footprint from electricity consumption, these data- centers can take up significant amounts of water, either directly for cooling or indirectly for electricity generation, bearing a larger than expected environmental footprint—in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=', it is estimated that 20% of datacenter servers’ direct water footprint is sourced from moderately to highly stressed watersheds and 50% of servers are at least partially supplied by power plants in water stressed areas [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In addition to the energy footprint datacenter/HPC operations, embodied carbon costs [8] such as those associated with manufacturing hardware for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' development and applications also matter, especially as hardware continues to advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As such, the environmental footprint of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' may go beyond the costs represented by carbon emissions of datacenters/HPCs alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 1: Modern AI’s Computational Demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Note the steep increase in just the past decade relative to the past 50 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Source: OpenAI & The Economist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As industry adoption and incorporation of algorithms into products and services become more commonplace, we have seen significant growth in both the amounts of training data and the size of the model itself [8] as the main means to realize performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Simultaneously, fundamental A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' research has continued to accomplish increasingly complex tasks with increasingly complex models and large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' These factors have, in turn, inevitably pushed similar growth trends in infrastructure investments required to keep pace with the increased amounts of training, inference, storage, and more arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='11581v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='AI] 27 Jan 2023 Deep and steep Computing power used in training Al systems Days spent calculating at one petaflop per second*, log scale 100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='4-month By fundamentals AlphaGoZerobecomes itsown doubling 10 teacher of the game Go O Language Speech O Vision 6 1 OGames 0 Other AlexNet, image classification with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='1 deep convolutional neural networks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='0001 Two-year doubling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content="00001 (Moore's Law) ←First era→ → Modern era 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='000001 Perceptron, a simple artificial neural network 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='0000001 1960 70 80 90 2000 10 20 Source:OpenAl 1petaflop=1015calculations(see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Coupled with the anticipated increase in internet traffic, consumer devices, and demand for the very products and services some of these algorithms support [5], these worrying trends in energy demand and its associated energy footprint are likely to only accelerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Even so, in the arms- race for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' superiority and operationalization, companies and institutions involved in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' research and its applications have continually expanded their datacenters and operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Google’s datacenter facilities span several countries while Meta has recently announced the construction of a new A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Research SuperCluster (RSC) claimed to be among the fastest and largest supercomputing centers upon completion [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In this race to construct an ever-increasing number (and size) of datacenters, supercomputing clusters, and supporting facilities, there are few signs that this race will slow down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Instead, com- panies are accepting this as an inevitability and are looking for ways to help offset their ever-increasing energy footprint, such as building their own additional energy production facilities to fuel their operations [10] [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Energy-efficient data infrastructure and green computing are hardly new concepts and have seen continued work and advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' From the development of efficient chips like Google’s TPUs [12] and other computing efficiency gains to the application of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' algorithms themselves to automate datacenter operations, there is a long list of existing prac- tices and current works-in-progress to address the energy- hungry and data-intensive appetite necessary to sustain these algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Though these advances in efficiency have kept pace with the increased computation/energy needs and offset demand thus far, there may be signs that this is unlikely to last [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' There also exists some debate on the true extent to which issues on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.’s sustainability and energy footprint are accurately described, largely driven by notable successes in realizing energy and computational efficiency in model training, datacenter/HPC operation, and hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' However, changes in climate resulting in rising temperatures and more extreme weather patterns are likely to stress cooling and already strained resources in many areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' While larger, well-equipped technology companies have the resources and incentives to act, develop, and adopt efficiently, there are still clear, unaddressed concerns if all A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' workflows move to the same hardware and software stack despite the efficiency benefits from centralization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As we run up against the limits of remaining efficiency gains, other ideas and implementations are needed, either as an anticipatory or preventative measure, in order to proactively develop strategies that bring the dis- course to these problems and their potential solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In the following sections, we discuss the prospects of encouraging energy efficiency across various levels of the research & development spectra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' and its applications: (1) the infrastructure and resource utilization level, (2) the individual user and behavioral level, and (3) the group and community of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' researchers and practitioners at large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' These three aspects cover issues from a micro-to-macro perspective but also emphasize a key point—no single change on any one level is likely to be as effective without corresponding changes on the other levels since these three aspects are part of a single whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' A concerted, unified effort is required in order to transition effectively to a greener ecosystem for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' research and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' To make our analyses more concrete in our discussions, we leverage data from the MIT SuperCloud [14], an operational peta-scale HPC system that is actively used for research, experimentation, and collaborations by the MIT research community in several disciplines across machine learning, deep learning, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' ENERGY & BEHAVIORAL CONSIDERATIONS In this section, we discuss potential improvements towards a more energy-aware compute and cluster optimization frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' While we discuss traditional aspects of datacenter/HPC management in reducing energy expenditure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' hardware, system-level), we also focus on non-traditional possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' We touch upon issues such as the economic considerations of energy consumption like the opportunity costs of energy purchases, the role/effect of user behavior in designing mecha- nisms to encourage energy-efficient behavior, changes in exist- ing behaviors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' from either the user side or datacenter/HPC management side), and combining—but balancing—existing energy saving mechanisms on the hardware/systems side with ones accounting for user behavior and incentives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' When it comes to energy efficiency, a simplified optimiza- tion framework is useful in understanding the objectives, the available choices/mechanisms are at our disposal to affect change, their dependence on one another, trade-offs, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' This way, we can simplify the overall optimization problem that operational datacenters/HPCs face: min qs,p,c E(qd, qs, p, c, ε) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' A(qd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' qs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' ε) ≥ α (1) where total energy expenditure E(·) and activity level A(·) of the datacenter/HPC can be affected by various factors: exam- ples include the “quantity” of compute resources demanded or currently utilized by users (qd) as well as their usage behaviors including but not limited to efficient/inefficient practices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' the “quantity” of compute resources supplied or available to users (qs) and associated resource settings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' the job scheduling system or resource allocation rule in place (p),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' control mechanisms (c) such as hardware settings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' power caps, clock rate settings) or other physical interventions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' rack placements, cooling setups) and “softer” mechanisms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' algorithmic, instrumentation) that may be in place, and ε which accounts for other factors such as temperature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=', ambient, distributions across racks, local climate) and others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' a datacenter’s fuel mix and energy purchasing patterns, maintenance schedules, electricity prices and energy mix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In other words, the goal is to minimize the energy ex- penditure E(·) of the datacenter subject to a constraint: the activity or performance level A(·) of the supercluster must be above some minimum, acceptable threshold α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' This constraint expresses a fundamental trade-off at the heart of energy- efficiency: reductions in energy consumption or expenditure need to be weighed against trade-offs in performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' jobs still need to be done at a reasonable pace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' If the performance level constraint α is not satisfied, attempts to reduce energy expenditure may produce perverse, unintended effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' for instance, if a change to reduce energy consumption results in noticeable performance degradation, then users may run more jobs for longer, producing the opposite effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Although one possibility is that higher throughput jobs can reduce total energy consumption by driving up power consumption but finishing in shorter periods as a result, we assume here that α corresponds to a bare minimum performance level—beneath which even these high throughput jobs contribute little to the overall energy footprint compared to the other kinds of workloads/operations present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Traditionally, resource management in datacenters/HPCs tends to take an approach closely aligned with the problem as outlined (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 1), minimizing energy expenditure primarily through three main ways: adjusting the available “supply” or amount of resources qs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' number/types of GPUs), adjusting resource allocation rules and schedulers p, and usage of control mechanisms c (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' hardware settings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' These mechanisms can be quite effective, cheap, and can easily produce intended results as they do not necessarily require coordination or know- how from users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' While much work has focused on optimiz- ing energy efficiency through these traditional mechanisms— affecting available compute resources, resource allocation and queuing/scheduling rules, or hardware/software and physical configurations [15] [16]—new sources of efficiency will likely need to be claimed from ε as we hit diminishing returns and, eventually, limits from traditional measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As easy sources of efficiency are exhausted, these limits will require looking beyond more traditional levers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=', p, qs, c) and towards less- traditional ones (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=', qd, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Energy, Power, & Opportunity Costs When considering the energy expenditure or carbon foot- print of HPCs/datacenters, what quantity should we focus on?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As framed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 1, the main objective E(·) can represent any number of quantities correlated with energy expenditure: kilowatt-hours, power usage effectiveness (PUE), pounds of CO2 emitted, amount of water used in cooling, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Besides these quantities, E(·) can also account for aspects like the fis- cal costs of the datacenter’s energy bill or even the opportunity costs of its choices, arising from the timing, the amounts, or the fuel composition of its energy demand and usage as well as how they affect the datacenter’s environmental footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The economic costs of a choice accounts not only for its direct fiscal or monetary costs, but also its opportunity costs—the cost of the best alternatives foregone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In this subsection, we discuss these opportunity costs and strategies to reduce these costs by changing energy purchasing behaviors like the timing of energy purchases and other usage patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' For instance, consider the usage patterns of the MIT Su- perCloud system [14] within a given year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Naturally, the demand and usage of the system’s overall resources will vary throughout a year, exhibiting regular patterns on different time scales within the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Just as demand and load vary, power 2 4 6 8 10 12 Month 200 250 300 350 400 450 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Power (kW) Power Consumption vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Sustainable Fuel Generation 5 6 7 8 % Total from Solar/Wind Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 2: Power Consumption vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Green Fuel Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Average monthly power consumption of MIT’s E1 hypercluster plotted against monthly average percentage of supplied total energy derived from solar and wind (2020-21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' There are potential opportunities—high power consumption when green energy production is low and vice versa instead of the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 2 4 6 8 10 12 Month 20 25 30 35 40 45 50 Real Time Avg Price ($/MWh) Energy Prices vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Sustainable Fuel Generation 5 6 7 8 % Total from Solar/Wind Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 3: Energy Prices vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Green Fuel Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Average monthly energy prices plotted against monthly average percentage of supplied total energy derived from solar and wind (2020-21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Prices are monthly locational marginal prices (LMP) from south eastern/central MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Note that energy prices tend to be lower when percentage of sustainable energy is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' consumption will also vary—more users and jobs generally translate into more computation and increased cooling costs, increasing power draw from existing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Beyond the dollars-and-cents of the HPC’s electricity bills, the make-up or composition of the energy supplied by the power company via the local grid can also influence the sustainability of a datacenter/HPC’s operations albeit in a less direct way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The different sources from which power is generated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' the fuel mix), supplied to, and consumed by the HPC carry an implicit environmental opportunity cost: the usage or purchase of power with a less sustainable fuel mix at a period in time forgoes usage of power generated with a greener fuel mix in that same time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' This, in turn, represents the foregone opportunity to offset some portion of existing energy expenditure while imposing an environment cost in the form of greater energy inefficiency as an externality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' One way to then improve energy efficiency is to shift energy expenditure more towards power sourced from higher ratios of sustainable fuel mixes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' generated with more sustainable sources like solar and wind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Figure 2 suggests there may be an opportunity to change the datacenter/HPC’s purchasing behavior for this strategy to be viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Over the course of the year, we see that the total share of fuel/energy produced from solar and wind is inversely related to the average amount of power used per month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The MIT Supercloud energy consumption has been relatively high when the share of renewable energy is low around June to August—similarly, energy consumption/expenditure is lower when the share of renewable energy in the fuel mix of the power supplied is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' One strategy to take advantage of this mis-match between power consumption and fuel mix, increase energy efficiency, and reduce the environmental opportunity cost is to purchase more power during times when sustainable energy takes up a larger share of the fuel mix (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' March to May) and either: (1) capitalize during that time period by encouraging more cluster utilization during those months or (2) store that energy to help offset energy consumption during times where the fuel mix is less sustainably sourced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Figure 3 suggests this strategy also carries financial ben- efits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' During springtime, from February to May, when the sustainable energy share of fuel mix tends to be high (> 8%), general energy prices tend to be extremely low ($20-$25 per megawatt-hour) and are some of the lowest prices of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' However, it is important to note that renewable energies like solar and wind may not always see stable generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' moreover, there are additional fixed costs incurred from setting up the relevant infrastructure that may be required in order to pursue strategies like the ones described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' We explore and discuss the application of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' to help stabilize sustainable energy generation as well as infrastructure investments as they relate to efficiency in the sections below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Temperature-aware & Weatherized Compute Optimization While changes in the regular, shorter-term behavior of datacenters/HPCs can be helpful, like those described above, longer-term structural changes and preparations are essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As changes in climate produce increasingly extreme weather events and rising temperatures [17], traditional mechanisms alone may be insufficient to brace for what is to come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In light of these upcoming challenges, energy-aware cluster optimization must find ways to explicitly account for factors in ε that, though difficult to anticipate, carry significant consequences to datacenter/HPC health and efficiency such as weather and climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' How would existing concepts and practices of cluster management and energy efficiency change with more extreme climate and more frequent weather events?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' What would weatherized compute optimization look like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 4, we see the monthly average temperature and its trend along with those of power consumption for the MIT Supercloud system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Throughout the year, there is a monotonic, one-to-one relationship between average monthly power consumption and average monthly (local) temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As temperatures become warmer heading into the spring and summer months, it takes more power to cool the facilities and maintain a sufficiently low temperature for normal oper- ations, resulting in increased power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' If average temperatures continue to climb even in the colder months as a consequence of climate change, cooling is likely to become more difficult and costly as previously efficient mechanisms for cooling facilities may suffer previously unseen stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 4: Power Consumption vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Green Fuel Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Average monthly power consumption of MIT Supercloud plotted against monthly aver- age temperature (in Fahrenheit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Note the near one-to-one relationship between temperature and power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As such, investments into infrastructure weatherization is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As changes in climate induce more extreme weather events and temperature ranges with increasing regularity, ex- isting methods to realize energy efficiency may no longer be as effective under more frequent or extreme weather/climate conditions especially if mechanisms only function effectively within a small band of temperature/climate conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Since historical data points of extreme weather can be rare (for now), a useful exercise can be a regularly conducted stress-test akin to the Dodd-Frank stress tests [18] enacted after the 2008 financial crisis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' these stress tests are conducted annually and provide simulated stress scenarios that test the resiliency of financial institutions in both its traditional functions/operations as well as with less traditional risks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' geopolitical, climate, infrastructure), helping identify areas in need of remediation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Similar stress scenarios and risk identification, conducted and evaluated regularly, for not just regular datacenter/HPC operations but also for climate and weather resiliency can help anticipate what energy efficiency (and inefficiency) looks like when considering future changes in weather and climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' For institutions with more than one HPC/datacenter, these exercises can provide opportunities to plan and coordinate across geo-scattered HPCs/datacenters to improve their col- lective resilience or develop re-routing backups in extreme weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Most importantly, these exercises can help anticipate and identify critical areas of infrastructure which require both a significant time and financial investment that may not come up otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Incentives, Behavior, & Mechanism Design Hardware and system-level mechanisms can carry much of the weight in producing energy savings under-the-hood and abstracting away difficulties without taking away from user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' If these interventions run into diminishing returns, then discovering remaining gains in efficiency will require work not only from the “supply” side of computing but also on the “demand” side, qd—the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Compared to the macro- level approach dealing with cluster/datacenter-wide hardware and system-level interventions, this micro-level approach can PowerConsumptionvs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='MonthlyTemperature 450 70 400 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='MonthlyTemperature (F) Power (kW) 350 60 300 50 250 40 200 2 4 6 8 10 12 Monthprovide additional flexibility but will require careful planning around mechanism design, user behavior, and user incentives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' From this perspective, the optimization problem faced by the datacenter changes from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 1 to min i ei(qd(i), qs, p, c, ε) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' ai(qd(i), qs, p, c, ε) ≥ αi ∀i where � i ei = E, � i ai = A (2) for each individual or representative user (or workload) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Whereas before the datacenter/HPC in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 1 had control mainly through qs, p, and c, now the main mechanism is through a specific user/profile/representative workload i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' This ultimately translates into the datacenter attempting to induce changes in the quantity of resources demanded qd, as reflected by qd(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Instead of total across-the-board quantities like total energy and total activity/performance, E(·) and A(·), we now focus on individual (or representative) users, profiles, or representative workloads and their energy usage and activity profiles, as denoted by ei(·), ai(·), and αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Naturally, by tailor- ing energy minimization efforts to representative user profiles and workloads, these mechanisms can reduce overall energy expenditure selectively in ways that systematic hardware inter- ventions cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' These micro-level approaches aim to induce behavioral changes in users through affecting incentives with the support of predictive analytics and instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' One example is the design of queues for finer user and workload segmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' these queues can improve job schedul- ing and execution using user-provided information (and other information) like the user’s stated preferences on energy efficiency, job urgency/patience, expected time completion, type of workload, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Policies can then be tailored more specifically with only the resources necessary, allowing for more efficient design elements by reducing idle time, over- allocation, and over-utilization of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' However, if queue selection and user intent conflict in situations where the user has an incentive towards a specific resource configuration different from the assigned one, this mechanism runs the risk of adverse selection—users mis-characterize their preferences and select themselves into queues where resources are fastest, most plentiful, or the most available, leaving select queues clogged and overtaxed and others largely, if not entirely, idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In the example above, too many self-characterizing choices are made available for users to potentially mis-represent their preferences and extract private benefits while imposing a social cost on the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' One alternative to balance these two factors of too much choice and too little control is to maintain a two-part mechanism: a fixed component that guarantees a specified minimum amount of energy efficiency and a variable component that allows for user choice to further scale energy efficient behavior, but only in certain respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' For instance, it has been shown that optimal GPU power-caps provide an effective way to control energy consumption with minimal impact on training speed [15] and user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' With these optimal power caps as the fixed base component, the variable component can be offered as a choice: if an user accepts increasingly stringent power caps on his/her allocated GPUs (or other restrictions), the user can then, in exchange, choose to have more GPUs allocated to his/her tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' These types of choice mechanisms require a cost-benefit analyses to balance individual net benefits/costs with system-level benefits/costs but can help induce energy-efficient changes in user behavior and computing demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Designing mechanisms can be difficult but predictive mod- els and analytical tools can help in understanding and evaluat- ing both utilization patterns as well as opportunities to affect them in an energy-efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Models that help forecast and relate energy prices, fuel mix, as well as energy expen- diture to one another can provide significant support in the decision-making process for optimizing energy purchases and consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Similarly, models leveraging data on compute demand and usage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' holidays, research deadlines) can help with scheduling, maintenance, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Though these mechanisms are not without their drawbacks, predictive analytics and in- strumentation can help mitigate these shortfalls by anticipating and analyzing behavior via data and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' CLIMATE-AWARE RESEARCH ECOSYSTEMS A significant part of the A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' research ecosystem is driven and structured by incentives to publish in notable, high- visibility conferences and journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' These venues serve as important forums for the A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' community—researchers, prac- titioners, and the state of research as a whole—to disseminate new and important findings, promote brands, seek/hire talent, highlight significant contributions and problems, exchange information, foster innovation and collaborative relationships, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' These contributions notwithstanding, the way the research ecosystem is currently structured can create incen- tives worth reconsidering when transitioning towards a more sustainable research environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As both fundamental research and applications in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' to various fields continue to grow, high-visibility venues will likely receive more focus and submissions as researchers and practitioners strive to publish in the “best” possible venue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Many metrics of success in fundamental and applied research are also heavily influenced, if not defined, by publishing in these venues—preferring or requiring that researchers, practitioners, and even job candidates to have publications at notable venues—which continues to serve as a common incentive and evaluative criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' With such a significant focus on publication in key conferences, how do these incentives drive the pattern of research activity and what environmental consequences do they carry, if any?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Previous works have studied the carbon footprint generated by participants traveling to conferences [19] [20] but less attention has focused on the effect of the distribution of deadlines themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Conferences deadlines are typically scattered throughout the year with each conference serving a specific domain or as a general purpose venue (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Specific dates are publicized several months ahead to give enough time for preparation and planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The distribution of these deadlines may induce certain patterns in aggregate research activity, compute demand, and therefore energy utilization, the last of which we use as a proxy for activity/demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As an exploratory analysis, we compare the number of conference deadlines per month from January 2020 to end of year 2021 with trends in monthly energy usage in the MIT Supercloud system (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' To help account for the confounding effects of seasonality, temperature, and other factors on energy utilization, we include data across two years (2021 & 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Given the way deadlines are structured, we might expect a lagging relationship where activity or compute demand, and hence energy utilization, might pick up in anticipation of upcoming deadlines—the larger the number or concentration of upcoming deadlines, the larger the increase in compute demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' As deadlines approach, users are accelerating their workloads, finishing or repeating experiments, and preparing for conference submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In Figure 5, we see some pick- up in energy usage leading up to the months with a high concentration of deadlines (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' July 2020)—such as the uptick starting around March/April 2020 and leading up to July 2020—but this may also be due to higher temperatures and cooling costs as noted earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' However, there is a sharper pickup in energy usage starting around Jan/Feb 2021 in anticipation of a notable concentration of deadlines in the subsequent months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' This sharp increase in energy usage is significantly higher than in the same period of the previous year despite no significant differences in average temperature or other known factors in those time periods between the two years—the only difference being the concentration/number of deadlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Overall, we also see that many deadlines tend to concentrate in the spring/summer across both years when the combination of higher temperatures and increased compute demand can exacerbate existing energy trends, resulting in significantly higher energy usage that taxes the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In the same period (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' the summer months), the fuel mix of the supplied power also has the lowest ratio of sustainable energy of the year, as seen earlier (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 2), which further contributes to an enlarged environmental footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' A natural question that may arise is: can we structure deadlines to spread out energy utilization and compute demand to benefit energy efficiency?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' If the same amount of compute is to be spent throughout an representative year of research activity regardless, then several options may help distribute that amount in a more sustainable fashion: (1) spread deadlines more uniformly throughout the year, (2) concentrate deadlines in the winter/spring months when preceding months are colder or see more sustainable fuel generation, or (3) abolish fixed deadlines in favor of rolling submissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Some venues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Transactions on Machine Learning Research) have already shifted to rolling submissions albeit for different reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' We note that our preliminary analysis is intentionally limited in scope as we focus exclusively on the MIT Supercloud system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Additionally, it neither accounts for other confounding factors explicitly nor does it show a definitive connection be- tween conference timings and usage/energy intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Rather, it is meant to bring attention to how structural incentives in the current A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' research ecosystem and community may not align optimally with desirable aspects of sustainability— with one example being conference deadlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' More work and data are required to tease out the full picture of the degree to which aggregate research activity and its energy footprint are affected by conference timings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' We hope that future work will undertake a finer analysis, accounting for details such as workload type, type of research activity represented, breakdown of activity and energy use by domain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' NLP), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' beyond just data from this cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' This requires more data, better data, data access, as well as willingness to share these data, which may not currently exist in sufficient amounts, a matter we discuss further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' TABLE I: List of notable conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The following con- ferences are considered for analysis (not exhaustive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Area/Discipline Conferences NLP/Speech EACL, InterSpeech, EMNLP, AKBC, ICASSP ISMIR, AACL-IJCNLP, COLING, CoNNL, WMT, EACL Computer Vision ICME, ICIP, SIGGRAPH, MIDL, ICCV, FG, ICMI, BMVC, WACV Robotics IROS, RRS, CoRL, ICRA General ML COLT, ICCC, ICPR, AAMAS, AISTATS, CHIL EMCL-PKDD, NeurIPS, ACML, AAAI, ICLR Data Mining SDM, KDD, SIGIR, RecSys, CIKM, ICDM WSDM, WWW Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' 5: Energy Usage vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Number of Conference Deadlines Average monthly power consumption of MIT’s E1 cluster plotted against number of monthly conference deadlines (Table I) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' CLIMATE-AWARE RESEARCH PRIORITIES A discussion on the sustainability of the current A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' re- search ecosystem and its incentives would be incomplete without discussing the thematic lines of work, both old and new, such an ecosystem should prioritize in order to improve its sustainability and keep its environmental footprint small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Novelty, Redundancies, & Efficiency Given the complexity and variety of research and appli- cations in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=', there are likely significant redundancies in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Many experiments usually begin with training Energy Usage vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='Conference Deadlines 700 EnergyUsage 6 600 5 Conference Deadlines (Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Power kW) 500 4 400 3 300 2 200 1 2020 6-2020 9-2020 2020 2020 2021 8-2021 10-2021 6known and proven models up to some pre-specified level of performance, depending on the research direction, before building atop these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Doing so may require some hyper- parameter search, if not full-blown optimization, resulting in multiple training runs and inevitably redundant runs, wasted compute, and additional energy costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Some redundancies can play a helpful role by training students and researchers when they start working on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' research where experience obtained from reproducing results can help shape best practices down the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' However, problems with reproducability of research only compound these redundancies as (multiple) attempts at replication also waste resources and energy when researchers and practitioners attempt to build off existing work or put previous work into practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' These difficulties in replicating published results are wide-spread and well-documented [21], resulting from inconsistent reporting of sensitivity to hyper- parameters and training settings (or complete lack thereof), poor communication, missed opportunities from reviewers, mis-representation, or some combination of the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In the ever-changing landscape of new research and model frameworks, problems with redundancy and reproducibility can carry additional implications for energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' If incentives to develop better performing models overshadow those for reproducibility and transparency, research efforts devoted to producing newer, better models will outpace efforts for clearer benchmarking and reporting, leaving transparency and resource efficiency efforts forever playing catch-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' For instance, when GPT-3 debuted, despite its impressive perfor- mance on generative language tasks, its training (not including experimentation during its development) was prohibitively costly and estimated at around $5 million using a specially designed supercomputer by Microsoft [22], making it very difficult for researchers to train and test on their own— only after its introduction, extensive usage, and popularization did work focus addressing its efficiency and other issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' safety, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' alignment, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Over-parameterization and big data may offer easy performance improvements, but an emphasis on jointly co-optimizing efficiency and performance in research may help avoid this efficiency-in-hindsight ap- proach and front-loading significant energy costs in model development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Some progress has been made in addressing these problems as Google, Meta, and other large players have highlighted best practices and standards that have helped to significantly reduce their own carbon footprints [23] [8] for state-of-the-art NLP models, such as efficient model selections and hardware/system choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Despite this, however, the fun- damental problem of information reporting and data availabil- ity still remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' To remedy this, there needs to be an active, systematic, and consistent approach towards collecting and reporting data/information (on energy usage, training settings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=') that incentivizes voluntary contribution and surveys a sufficiently broad swath of sources to be representative of the diversity of workloads in research and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Measurement, Reporting, & Transparency Various works have produced estimates in attempts to quantify the carbon or energy footprint of deep learning model training with estimates ranging from as high as 5x the average lifetime emissions of a car [24] to as low as 10−5 times that amount [23] for state-of-the-art transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' These estimates are inherently variable and difficult—not only due to differences in aspects like hardware (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' GPU vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' TPU)— in both the approach taken to quantify these costs and their resulting accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' These difficulties in accurate estimation highlight the importance of regularly detailing energy usage and other information in research alongside typical items like performance results and ablation tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Moreover, while many estimates have focused on training costs, even less clear are the costs arising through a model’s entire life-cycle, which are particularly important in industry and applied settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Even so, there exist even less data on the costs of inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The discrepancies in, and even availability of, these esti- mates can be due to several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The first is resource asymmetry—not only do different companies, groups, and individuals have different amounts of computational resources, they also have different computational setups so certain met- rics and calculations may naturally vary depending on the underlying technological stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' This differentiation similarly applies in academic disciplines where a base model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' graph neural networks) may branch out into highly special- ized, differentiated variants depending on the field or task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' social networks vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' molecular predictions), resulting in significantly different training procedures, learning dynam- ics, energy footprints, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Different needs, resources, and constraints largely determine variations across research and development workflows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' as such, when a company or institution reports realized gains in efficiency or savings, these gains may only be realizable on their systems, with their resources/hardware/configuration, or limited to a specific class of models that are reported by, or essential to, said organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Though a seemingly simple solution would be to move over to services provided by organizations with the hardware and technical capabilities to realize such efficiencies, there are ethical concerns and market concentration issues that require addressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Even with similar tasks across companies and industries, different domains are also characterized by other considerations and constraints such as the lack of tech- nical expertise, specific resource and regulatory constraints, and other requirements like model privacy or interpretability that may outweigh model performance and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' At its worst, resource asymmetries can hamper reproducibility and verification efforts: if state-of-the-art models developed by large, well-equipped research groups are too costly and resource-intensive to train for others, how can their results and estimates be reproduced or verified?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Along with the resource asymmetry, information asymmetry can discourage and dis-incentivize researchers and practition- ers from reporting necessary or relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Some examples of these asymmetries, besides ones mentioned earlier like inconsistent reporting of training settings as well as poor communication and presentation of research results, can arise in part from incentives to preserve competitive advantages and other sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Incentives to protect and preserve a competitive edge from peers and competitors can discourage full, transparent reporting of information especially if these models and research tie into a company’s products and services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Even when reporting, these incentives may limit the amount of information made available to the wider research community, leading to confusion around estimates and methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Incentives to keep information, and its benefits, private for competitive advantage can lead to con- tinued information asymmetries in a self-reinforcing cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Voluntary reporting may then be dominated by larger, better- equipped groups with the resources and technical ability to optimize their operations which, though well-intentioned, will likely not reflect the true extent of the overall, or even the average, environmental footprint of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Moreover, despite the focus on the footprint and costs of training, data and estimates on inference are even scarcer despite its significance—the few estimates, where available, put inference at 90% of production ML infrastructure costs [25] and 80%-90% of energy costs [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' While training enjoys scaling benefits that saturate GPUs, the different performance requirements of inference can result in poor GPU utilization since inference queries are unable to realize the parallelism that offline mini-batch training enjoys [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Low resource- efficiency and utilization is quite common: AWS reports p3 GPU instances at only 10%-30% utilization [25] and even Google’s TPUs exhibit a utilization of 28% on average [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The issues outlined above all point to a common set of problems that require (1) a better, more representative idea of the kind of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' models, and the underlying resources, used across disciplines, domains, and communities, (2) a common set of meaningful metrics, and (3) incentives through both existing avenues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' conferences, papers) and new ones such as forums, competitions, leaderboards, or open challenges to encourage reporting of energy/utilization data and develop- ment of more energy-efficient models rather than just better performing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' To accurately quantify the environmental footprint, it is essential to capture costs with metrics that realistically reflect and represent the workloads undertaken in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' research and practice—as well as the burdens and en- ergy footprint associated with state-of-the-art models on more representative computational setups rather than in the most efficient, advanced settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' To incentivize consistent reporting and sharing of data, the research community needs forums that prioritize energy-efficient models and methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' For instance, a Green A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' challenge (in development) that aims to cast the problem explicitly by challenging participants to max- imize performance given explicit training and energy budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Lastly, facilities should also provide the central infrastructure, user interfaces, and analytical tools/instrumentation/logging to further encourage easy reporting and sharing of data, especially since not all users are equipped with the expertise to manually report relevant data and information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' for Energy Savings, Generation, & Discovery Despite its potential environmental footprint, some of the most impressive applications of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' algorithms have included ones that help generate energy savings themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' One exam- ple has been Google and DeepMind’s use of neural networks to monitor and optimize their datacenters, reducing the amount of energy spent for cooling by 40% and PUE by 15% in live tests [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Similar examples abound, but beyond energy savings, continued and improved sustainability will also require work from the other side of the equation: energy generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The study and application of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' to energy discovery and generation should be strongly incentivized given its immediate benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Current examples include the application of algo- rithms to stabilize and boost sustainable energy generation: wind farms provide inexpensive, carbon-free energy but can be unpredictable, making planning and energy delivery/storage difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' In response, DeepMind has developed neural net- works trained on weather forecasts and historical turbine data to forecast energy output 36 hours ahead, making early recommendations on optimal hourly delivery commitments to the grid possible [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Beyond existing energy sources, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' research can help push forward new sustainable energy sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Recent work has shown how deep reinforcement learning can help control nuclear fusion [31] by learning to control and change the shape of plasma via manipulation of its magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Scientific collaborations, especially as they relate to development of new energy sources or improvements in existing energy generation, should receive equal priority and recognition as state-of-the-art performance improvements in areas like vision and NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' To do so, partnerships with scientific and energy researchers should be encouraged and made more accessible to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' researchers and practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Similarly, benchmark energy datasets should be constructed and made easily accessible just like standard data benchmarks in NLP and vision—moreover, these energy datasets should receive continuous updates and testing due to the inherently variable behavior of wind, weather, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' CONCLUSION There are many dimensions of this multi-faceted problem that are not addressed in this paper due to space limitations but are important for consideration nonetheless such as the equity and accessibility aspects of energy-efficient computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Though daunting, we hope our discussions of these problems and their potential solutions will provide a framework that spurs further discussion, and most importantly action, on these various issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' ACKNOWLEDGMENT The authors acknowledge the MIT SuperCloud [14] and Lincoln Laboratory Supercomputing Center for providing HPC and consultation resources that have contributed to the research results reported within this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The authors acknowledge the MIT SuperCloud team: William Arcand, William Berg- eron, Chansup Byun, Michael Houle, Jeremy Kepner, Anna Klein, Peter Michaleas, Lauren Milechin, Julie Mullen, Albert Reuther, Antonio Rosa, and Charles Yee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' The authors also wish to acknowledge the following individuals for their con- tributions and support: Bob Bond, Allan Vanterpool, Tucker Hamilton, Jeff Gottschalk, Tim Kraska, Mike Kanaan, Charles Leiserson, Dave Martinez, John Radovan, Steve Rejto, Daniela Rus, Marc Zissman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' REFERENCES [1] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFJT4oBgHgl3EQfkyzq/content/2301.11581v1.pdf'} +page_content=' Environmental Protection Agency, “Green power partnership national top 100,” 2022.' metadata={'source': 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--- /dev/null +++ b/6NE3T4oBgHgl3EQfpgqP/content/tmp_files/2301.04643v1.pdf.txt @@ -0,0 +1,794 @@ +tieval: AN EVALUATION FRAMEWORK FOR +TEMPORAL INFORMATION EXTRACTION SYSTEMS +Hugo Sousa +1,2, Alípio Jorge +1,2, and Ricardo Campos +1,3,4 +1INESC TEC, Portugal +2University of Porto, Portugal +3Polytechnic Institute of Tomar, Portugal +4Ci2 - Smart Cities Research Center, Portugal +{hugo.o.sousa, alipio.jorge, ricardo.campos}@inesctec.pt +January 12, 2023 +ABSTRACT +Temporal information extraction (TIE) has attracted a great deal of interest over the last two decades, +leading to the development of a significant number of datasets. Despite its benefits, having access to +a large volume of corpora makes it difficult when it comes to benchmark TIE systems. On the one +hand, different datasets have different annotation schemes, thus hindering the comparison between +competitors across different corpora. On the other hand, the fact that each corpus is commonly +disseminated in a different format requires a considerable engineering effort for a researcher/prac- +titioner to develop parsers for all of them. This constraint forces researchers to select a limited +amount of datasets to evaluate their systems which consequently limits the comparability of the +systems. Yet another obstacle that hinders the comparability of the TIE systems is the evaluation +metric employed. While most research works adopt traditional metrics such as precision, recall, and +F1, a few others prefer temporal awareness – a metric tailored to be more comprehensive on the +evaluation of temporal systems. Although the reason for the absence of temporal awareness in the +evaluation of most systems is not clear, one of the factors that certainly weights this decision is the +necessity to implement the temporal closure algorithm in order to compute temporal awareness, which +is not straightforward to implement neither is currently easily available. All in all, these problems +have limited the fair comparison between approaches and consequently, the development of temporal +extraction systems. To mitigate these problems, we have developed tieval, a Python library that +provides a concise interface for importing different corpora and is equipped with domain-specific +operations that facilitate system evaluation. In this paper, we present the first public release of tieval +and highlight its most relevant features. The library is available as open source, under MIT License, +at PyPI1 and GitHub2. +Figure 1: tieval logo. +1https://pypi.org/project/tieval/ +2https://github.com/LIAAD/tieval +arXiv:2301.04643v1 [cs.CL] 11 Jan 2023 + +ti +Valtieval +1 +Introduction +Understanding the temporal order of events is essential to human communication. We, humans, can easily understand +the relative order of events in a conversation or when reading a news article. However, many challenges are raised when +we try to automate such tasks with a computer program. The first difficulty that emerges is how to represent temporal +information. Since in most cases we do not explicitly specify the start and end time of each event, temporal information, +such as order and time span, ends up being inferred from the events themselves. To this regard, computer algorithms +can make use of temporal clues in the text, and of external sources, such as knowledge-bases, to anchor events on a +timeline. For instance, in the sentence “We went to dinner after the game.”, two events, “dinner” and “game”, can be +automatically identified and used, despite the lack of explicit temporal information, to recreate a timeline of events +(see Figure 2) supported on the word “after”. The ordering of events and the knowledge about them, can be further +expanded if used together with appropriate external sources. For instance, the event “game” can be contextualized +and anchored on the timeline by searching for information on a knowledge-base. However, in the case of the “dinner” +event, it turns out impossible to know the exact time of occurrence unless it is specified in the text. This shows that +representing temporal information is not a trivial task, since there are several borderline cases for which no standard +approach has been established. +Figure 2: Relative timeline of events that can be inferred from the running example. +Over the years, and particularly in the last two decades, this problem has been highly studied, leading to several +proposals from the research community Campos et al. [2014], Leeuwenberg and Moens [2019]. Most of the proposals +were in the origin of the emergence of different annotation schemes and the various corpora that we have today at our +disposal Naik et al. [2019], Ning et al. [2018a], UzZaman et al. [2013]. Although these efforts have been essential +to mature temporal information extraction and its subtasks – such as temporal expression identification or temporal +relation classification – they also pose some problems upon the process of benchmarking different methods. One of the +problems has its roots in the fact that evaluating the methods, often requires reading multiple corpora, each of which has +a different perspective on temporal representation, ultimately preventing comparability among the different methods +and corpora. This is compounded by the fact that corpora are stored in a variety of formats (e.g., XML, TimeML, or +table ), which requires a considerable engineering effort to load them all. +Another issue that limits the comparison between systems is the lack of standardization in the metrics used in the +evaluation process. This is a particular problem of temporal relation extraction – a subtask of TIE, which deals with the +identification and classification of the temporal relations between entities – where different metrics are often employed +during the evaluation process. While initially systems were evaluated and compared using standard metrics, such as +recall, precision, and F-score Verhagen et al. [2007, 2010], more recently, metrics such as temporal awareness UzZaman +and Allen [2011] have proven to be more reliable in the evaluation of temporal relation extraction methods. The +reasoning behind this is that, while traditional metrics focus on the local effectiveness of the model, temporal awareness +better understands the relative order of events by considering the global temporal structure of the predictions. This is +accomplished by taking into account the temporal relations that can be inferred from the established ones (a process +typically referred to as temporal closure), making this a more comprehensive metric for evaluating temporal systems. +Despite the emergence of this temporal awareness, many studies still rely solely on traditional metrics to evaluate +their system. We speculate that this is due to the fact that temporal awareness requires domain-specific operations +such as temporal closure – which are not (yet) readily available in every framework and therefore require individual +implementation by each research group. In addition, temporal awareness requires the implementation of a strategy to +deal with inconsistent predictions of the system, which is generally not explored in recent studies. +To mitigate the above issues, we developed tieval, a Python library that enables the development and evaluation of +TIE systems. This framework provides a simple interface to download and read TIE corpora in various formats. It +currently covers well-known corpus – such as TempEval-3 UzZaman et al. [2013], TDDiscource Naik et al. [2019], and +MeanTime Minard et al. [2016] – however, it lays the foundations for others to be included by providing base classes +for the construction of the corpus. It also provides domain-specific operations – such as temporal closure and simple +translation of intervals into point relations – that can be used to develop TIE systems. In addition to this, it includes an +evaluation infrastructure for a comprehensive assessment of the effectiveness of the different models being evaluated. +Because tieval supports the entire development pipeline of TIE, it can also be used to ensure reproducibility and fair +benchmarking of future research. The main contributions of tieval are the following: +2 + +tieval +1. it gathers the multiple corpora for the development of TIE systems, making it easy to access with just a few +lines of code; +2. it facilitates access to domain-specific operations, such as interval to point relation and temporal closure, as +well as metrics such as temporal awareness; +3. it provides a standard framework, thus making it easy for new methods to be compared against previous ones. +The remaining of the paper is organized as follows: The next section, provides an overview of recent work in TIE and +some of its software. We then proceed to present the tieval package in section 3. We start with a general introduction +and then go into some of its most relevant features. Section 5 serves to present our thoughts on what we strive to be +next steps in the development of the framework. +2 +Related Work +Extracting temporal information from documents written in natural language in an inter-operable format has long +been an interest of the artificial intelligence community Ling and Weld [2010], Derczynski et al. [2015]. Since the +introduction of the Time Markup Language (TimeML) Pustejovsky et al. [2003a], in 2003, the temporal graph has +become the de-facto standard to represent temporal information. In this graph, the nodes are temporal entities and the +edges are the temporal relation that hold between them. The temporal entities can take two forms: event expressions, +which are defined as situations that happened (e.g., “went” or “bought”); and temporal expressions (timex), which can +convey temporal information explicitly (e.g., “October 27, 199”) or implicitly (e.g., “a few years ago”) Campos et al. +[2017]. The temporal relations are held in the form of temporal links (tlink) that contain temporal relations between +pairs of events (E-E relations), events and time expressions (E-T relations), and events and document creation time +(E-DCT relations), where DCT is a special timex that stores document creation time. Overall, these temporal relations +can take thirteen types, which is the number of relations that can exist between two time intervals Allen [1983]. +The first corpus that was annotated with this scheme was TimeBank Pustejovsky et al. [2003b]. The release of this +corpus, dated from 2003, sparked a wave of research in the field later on also used on the TempEval shared tasks +UzZaman et al. [2013], Verhagen et al. [2007, 2010]. These tasks end up segmenting TIE into a set of sub-problems +that can be conceptually defined as temporal entity identification, tlink identification, and tlink classification. Although +some works developed systems for more than one of these sub-tasks, most of the systems are concerned with only one +of them. Furthermore, temporal entity identification systems are traditionally partitioned into subsystems for the several +classes of temporal entities. For example, for the TimeBank corpus, one system is usually trained to identify events and +another to identify timexs. The tieval architecture follows this natural decomposition of the TIE. +The TimeBank corpus, and more abstractly, the TimeML annotation scheme was widely studied by the community. +Such scrutiny lead to the emergence of several new corpora. Some used the TimeML annotation scheme to create +new corpora, such as AQUAINT Graff [2002] and the Platinum corpus UzZaman et al. [2013], while others were +concerned in extending the annotation scheme to other languages. The most remarkable effort on this domain was +the TempEval-2 shared task Verhagen et al. [2010] that produced corpora for Chinese Li et al. [2014], French Bittar +et al. [2011], Italian Caselli et al. [2011], and the Spanish Nieto and Saurí [2012] language. Another noteworthy effort +is the MeanTime corpus Minard et al. [2016] in which the authors annotated 120 news articles written in English +from Wikinews3, and translated the texts into Italian, Spanish, and Dutch. Costa and Branco Costa and Branco [2012] +followed a similar process to construct TimeBankPT, translating the original TimeBank to Portuguese and adapting +the annotations when needed. Apart from the extensions to other languages, the TimeML annotation scheme was also +extended to other domains. A concrete example is the case of the clinical domain for which two corpora have been +produced, the i2b2 Sun et al. [2013] and THYME Styler IV et al. [2014]4. Further significant contributions were the +proposals that explored ways to mitigate some of the issues found on the TimeBank annotation effort, such as: sparse +annotation – TimeBank-Dense Cassidy et al. [2014] and TDDiscourse Naik et al. [2019]; improve inter-annotator +agreement – MATRES Ning et al. [2018a]; and include other sources of knowledge – TCR Ning et al. [2018b] and +RED O’Gorman et al. [2016]. +Aside from the TimeML, and related approaches, there have also been other proposals that were explored by the +research community. One of them is absolute timeline placement, in which the temporal entities are directly anchored +on a timeline by labeling each entity with the time (or time span) of occurrence. The most remarkable efforts in this +direction were produced by Reimers et al. Reimers et al. [2016] – which produced the EventTime corpus by annotating +the events in TimeBank with a specific day, or span of days – and Leeuwenberg and Moens Leeuwenberg and Moens +3https://en.wikinews.org/ +4These corpora are not available for open access and, as a consequence, we were not able to include them on the framework. +3 + +tieval +[2020] – which annotated 169 clinical records from the i2b2 corpus with the most likely start and end time of each +event along with a lower and upper bound. +This shows that several corpora have been introduced for the TIE task. However, the fact that they were released in +different formats makes it hard to leverage their power, which is one of the issues mitigated by tieval. +To the best of our knowledge, the only framework that made available TIE operations – including temporal closure +and temporal awareness – is the Anafora Tools project5 which was built to work with files stored in the Anafora XML +format Chen and Styler [2013], used to annotate the THYME corpus Styler IV et al. [2014]. The framework presented +in this paper aims to be a more general tool, unifying all corpora in a single format. +3 +tieval +Our vision for tieval was to build a framework that would support and facilitate the evaluation of TIE systems. With +the development of libraries such as Numpy, TensorFlow, and PyTorch, Python has established itself as the programming +language of choice within the machine learning community. For that reason, we built tieval in Python. To facilitate +the installation we made it available on Python Package Index (PyPI)6. Thus, the toolkit can be easily installed through +pip, as follows: +$ pip install tieval==0.0.6 +In this paper, we will use version 0.0.6, which is the first and the most recent version of the package. However, the +reader is advised to install the newest release at the time of reading the paper and refer to the project repository for +up-to-date documentation. Furthermore, for users that might be interested in contributing to tieval, we encourage +forking the source repository and making a pull request. +tieval contains three modules that represent the three cornerstones of any machine learning project: datasets, +models, and evaluation. The datasets module is responsible for downloading and reading the corpora available for +TIE, the models module is responsible for the construction of the models, and the evaluation module has methods to +make a proper evaluation for each of the TIE tasks. In the following sections, we will present the inner workings of the +framework with scripts to exemplify the usability of the framework. +3.1 +Datasets +With tieval, we wanted to mitigate the issues referred above by making it easy for the user to work with several +corpora with a few lines of code. To that end, we developed an architecture that would unify the different annotations +and storing formats of the corpus. This architecture is composed of several objects which are depicted in Figure 3. +Figure 3: Objects used to represent a dataset on tieval. The arrow represent a relation of “Iterable”. +The Dataset object is the final representation of each corpus. It compiles the set of all the documents in the corpus on +the documents attribute which is segmented into the train and test attributes whenever provided in the original paper7. +Each document is then stored as an instance of the Document class (see the Document grey box in Figure 3), which +contains all the information necessary for TIE, more specifically: +name a string that contains the name of the document (e.g. “wsj_0026.tml”); +5https://github.com/bethard/anaforatools +6https://pypi.org/project/tieval/ +7When no standard train/test split is provided by the authors all the documents are placed on the train attribute. +4 + +Dataset +Document +Entity +TLink +.documents +.name +.text +.source +.train +.text +.value +.target +.test +.dct +.endpoints +.relation +.entities +**kwargs +**kwargs +.tlinkstieval +text a string with the raw text of the document; +dct is a Timex that contains the document creation time (e.g. Timex("12-10-2004")); +entities is the set of Entities – either a Timex or Event – that are annotated on the corpus. Each Entity is, at is core, a +data class made to store all the info provided on the annotation. Therefore, it has to be flexible to accommodate +for the different types of information provided in different corpus. For instances, the GraphEve corpus provides +the lemma for each event while TempEval-2 does not; +tlinks a set o TLink’s that stores the temporal relations annotated on the document. Each TLink contains a source and +target entity as well as the temporal relation between them – on the relation attribute. +A special remark needs to be made about the relation attribute of the TLink object. When initiating a TLink instance +one needs to pass the temporal relation that holds between the two temporal entities (the source and the target). In +most of the corpora this is one of the thirteen temporal relations Allen [1983] that can hold between two time intervals, +however, there are corpora where the annotators were more flexible on the type of relations. Examples of this are the +TempEval-2 and the MATRES corpus. On TempEval-2 the annotators were allowed to give more ambiguous relations as +“BEFORE-OR-OVERLAP” and “OVERLAP-OR-AFTER”. In MATRES the annotators were asked to provide the temporal +relation between the start points of the temporal entities. In order to accommodate the several types of annotations, we +build TemporalRelation object, which handles the relation that was annotated. Inside this object, every relation is +represented in point relations – instead of the traditional interval relations. Figure 4 shows how to represent the interval +relation “BEFORE” into a point relation. A relative relation is also included in the figure for illustrative purposes. +Figure 4: Relative timeline of events that can be inferred from the running example. +Note that the “BEFORE-OR-OVERLAP” relation on TempEval-2 represents an uncertainty of the annotator between the +end time of the source entity and the start time of the target entity, however, the annotator is certain about the remaining +point relations. Further note that, although we explicitly state four-point relations in Figure 4, upon the adaptation of +the current datasets into tieval format, three of them are redundant, as the point relation “end A < start B” completely +defines the remaining point relations. Therefore, on tieval, whenever there is a new dataset to include, the user can +provide the relation in the way that is most appropriate, as shown in Listing 1. +Listing 1: Different ways to pass the temporal relation to the TLink object. The first argument (X) is the source entity, +the second (Y) is the target entity, and the third is the temporal relation between them. This can be passed as an interval +relation, “before”, or as a point relation, in the form of a dictionary structure. On the latter, the interpretation for the +expected keys is the following: “x” and “y” stands for the source and target entity, respectively; while the “s” and “e” +stand for “start” and “end”. As an example, “xe_ys” is the point relation between the source end and the target start. +from tieval.links import TLink +tl1 = TLink("X", "Y", "before") +tl2 = TLink("X", "Y", {"xe_ys": "<"}) +tl3 = TLink("X", "Y", {"xs_ys": "<", "xs_ye": "<", "xe_ys": "<", "xe_ye": "<" ,}) +In order to reach a standardized representation for the different corpora, we developed a reader for each of the +corpus. Each dataset reader has inherited from an abstract base class, named BaseDocumentReader, which requires the +implementation of five methods named after the five attributes used to create an instance of a Document: name, text, dct, +entities, and tlinks. To extract this information, the base class contains three attributes: the path for the document being +read; the content of the dictionary produced by parsing the document with the xmltodict8 library; and the xml attribute +that results from parsing the file with the xml9 library. Note that, while json is nowadays the standard format for the +8https://pypi.org/project/xmltodict/ +9https://docs.python.org/3/library/xml.etree.elementtree.html +5 + +Relative Relation +Interval Relation +Point Relationtieval +exchange of the information, we had to resort to xml as most datasets were stored in that format. The script presented in +Listing 2 illustrates how to read a document from the TempEval-3 corpus with the TempEval3DocumentReader. +Listing 2: Read a document of the TempEval-3 corpus. +from tieval import datasets +path = "tempeval -3/ wsj_0026.tml" +reader = datasets.TempEval3DocumentReader(path) +doc = reader.read() +To fully integrate a new corpus on the library – and automatically read the entire corpus – the user just needs to add +an entry on the DATASETS_METADATA dictionary with the metadata necessary to read the document. This information +will be used on the read function of the datasets module, which only requires the name of the corpus to produce +an instance of the Dataset object with all the annotations provided in there. The script in Listing 3 presents how to +perform such operation. +Listing 3: Read the TempEval-3 corpus. +from tieval import datasets +te3 = datasets.read("TempEval -3") +The current version of tieval natively supports the download and reading of an extensive list of corpora for TIE. A +complete list of the corpora considered is provided in Table 1. In order to ensure long-term support for these corpora, +we created a repository with them. Besides that, it also has the advantage that we can standardize the structure of the +folders and add useful information to the raw datasets (for instance, the spans of the temporal entities identified on the +text) and fix errors on the original annotation10. For that reason, we were careful to verify the license for each of the +corpora and publish only the ones that allowed for redistribution or did not provide any license. +Table 1: The corpora currently supported on tieval. +Language +# Docs +# Events +# Timexs +# Tlinks +AncientTimes +Arabic +5 +0 +106 +0 +Dutch +5 +0 +130 +0 +English +5 +0 +311 +0 +French +5 +0 +290 +0 +German +5 +0 +196 +0 +Italian +5 +0 +234 +0 +Spanish +5 +0 +217 +0 +Vietnamese +4 +0 +120 +0 +Aquaint +English +72 +4,351 +639 +5,832 +EventTime +English +36 +1,498 +0 +0 +GraphEVE +English +103 +4,298 +0 +18,204 +KRAUTS +German +192 +0 +1,282 +0 +MATRES +English +274 +6,065 +0 +13,504 +MeanTime +English +120 +1,882 +349 +1,753 +Spanish +120 +2,000 +344 +1,975 +Dutch +120 +1,346 +346 +1,487 +Italian +120 +1,980 +338 +1,675 +Narrative Container +English +63 +3,559 +439 +737 +Continued on next page +10The changes made on the original corpus are detailed on the file logbook.rst in the docs folder of the project repository. +6 + +tieval +Table 1: The corpora currently supported on tieval. (Continued) +Professor Heideltime +English +24,642 +0 +254,803 +0 +French +27,154 +0 +83,431 +0 +German +19,095 +0 +194,043 +0 +Italian +9,619 +0 +58,823 +0 +Portuguese +24,293 +0 +111,810 +0 +Spanish +33,266 +0 +348,011 +0 +Platinum (TempEval-3) +English +20 +748 +158 +929 +TimeBank +Spanish +210 +12,384 +1,532 +21,107 +French +108 +2,115 +533 +2,303 +Portuguese +182 +7,887 +1,409 +6,538 +English +183 +6,681 +1,426 +5,120 +TimeBank 1.2 +English +183 +7,940 +1,414 +6,413 +TCR +English +25 +1,134 +242 +3,515 +TDDiscourse +English +34 +1,101 +0 +6,150 +TempEval 2 +Chinese +52 +4,783 +946 +7,802 +English +182 +6,656 +1,390 +5,945 +French +83 +1,301 +367 +372 +Italian +64 +5,377 +653 +6,884 +Korean +18 +2,583 +317 +0 +Spanish +210 +12,384 +1,502 +13,304 +English +275 +11,780 +2,223 +11,881 +TimeBank Dense +English +36 +1,712 +289 +12,715 +TrainT3 (TempEval-3) +Spanish +175 +10,686 +1,269 +17,283 +Wikiwars +English +22 +0 +2,662 +0 +German +22 +0 +2,239 +0 +3.2 +Models +The current version of tieval has four built-in models, namely: a baseline for timex identification; the HeidelTime +model Strötgen et al. [2013] for timex identification and classification; a baseline for event identification; and the +CogCompTime 2.0 model Ning et al. [2019] for tlink classification. The availability of these four models is intended for +practitioners that may want to experiment using any layer of temporal information in their specific application. Apart +from that, it also provide researchers the implementation of baseline models for reference in their work. +For the baseline models, we provide pre-trained weights, however, the user can also train the model from scratch. A +description of each of the models is provided below: +TimexIdentificationBaseline For this baseline we trained – from scratch – the spaCy11 named entity recognition +model to identify the timexs on the TempEval-3 corpus. +EventIdentificationBaseline This model has the same architecture of the TimexIdentificationBaseline but was +trained to identify events rather than timexs on the TempEval-3 corpus. +HeidelTime This model is a widely recognized multilingual temporal tagger which was original written in Java12. +However there have been efforts to build python wrappers. In tieval we used the py_heideltime wrapper +which is available on GitHub13. +11https://spacy.io/ +12https://github.com/HeidelTime/heideltime +13https://github.com/hmosousa/py_heideltime +7 + +tieval +CogCompTime2 This model leverages the ELMo Peters et al. [2018] word embeddings and the TempProb Ning et al. +[2018c] knowledge base to classify the temporal relation between a pair of temporal entities Ning et al. [2019]. +Our implementation was adapted from the repository made available14 by the authors. +Listing 4 presents a script that would download the baseline model for temporal expression identification +(TimexIdentificationBaseline), train the model on the TempEval-3 train set, and produce predictions for the +TempEval-3 test set. +Listing 4: How to download, train, and predict with for the temporal identification task. +from tieval import models +model = model.TimexIdentificationBaseline () +model.fit(te3.train) +predictions = model.predict(te3.test) +A user interested in releasing his/her model in tieval can do it by creating a subclass of one of our base classes for +models. There are two base classes: a BaseModel which just requires the implementation of the predict method +which is intended for models that are available in other repositories – for instance, the HeidelTime model – and a +BaseTrainableModel which, besides the predict, requires the implementation of the fit method, which implements +the training loop for the model. +3.3 +Evaluation +tieval provides an evaluation function for four subtasks of TIE, more specifically: timex identification, event +identification, tlink identification, and tlink classification. +Table 2: The results obtained by evaluating the four models integrated in tieval on the Platinum (TempEval-3 test set), +TCR, and MeanTime (the English version) corpus. P stands for precision, R for recall, F1 is the F1-score, and TF1 is +the temporal awareness. All the results in the table are micro metrics. +Platinum +TCR +MeanTime +P +R +F1 (TF1) +P +R +F1 (TF1) +P +R +F1 (TF1) +TimexBaseline +88.1 +75.4 +81.2 +75.4 +82.0 +78.6 +23.7 +57.1 +33.5 +HeidelTime +84.0 +79.4 +81.8 +70.6 +80.6 +75.3 +26.5 +65.8 +37.8 +EventBaseline +74.6 +80.5 +77.5 +48.3 +92.6 +63.5 +25.8 +54.1 +34.9 +CogCompTime2 +39.7 +39.7 +39.7 (39.3) +75.4 +75.4 +75.4 (69.3) +30.7 +28.6 +29.6 (28.9) +The input is standard for all the evaluation functions: annotations, a dictionary with the name of the documents +as keys and the annotations as values; predictions, follows the same structure of the annotations but for each +document key contains the predictions made by a model. The output of the functions is dependent on the task being +evaluated. For the identification tasks (timex, event, and tlink) the function produces the standard macro and micro +metrics for precision, recall, and f1-score. Listing 5 presents a script that evaluates the predictions made by the event +baseline model in the TempEval-3 test set. +Listing 5: Evaluate event baseline model on the TempEval-3 test set. +from tieval import evaluate +annotations = {doc.name: doc.events for doc in te3.test} +result = evaluate.event_identification(annotations , predictions) +Table 2 depicts the results obtained by the implemented on benchmark corpora. Note that TF1 is the temproal awareness +metric and is only computed for CogCompTime2 (the only tlink classification system). Another interesting remark is +the fact that the TimexBaseline achieves effectiveness comparable to HeidelTime despite its simplicity. +The tlink classification is the most elaborate evaluator as it also computes the temporal awareness metric UzZaman and +Allen [2011]. The complexity of the calculation of temporal awareness lies in the computation of temporal closure. With +temporal_closure the closure operation can be easily performed on the document level, with the closure method of +14https://github.com/qiangning/NeuralTemporalRelation-EMNLP19 +8 + +tieval +the Document object, or applied to a set of tlink’s with the temporal_closure function available on the library. The +script in Listing 6 illustrates how to perform such operations. +Listing 6: How to compute the temporal closure with a Document object and with a set of TLink’s. +from tieval import temporal_closure +doc = te3["wsj_0026.tml"] +closure_tlinks = doc.closure +closure_tlinks = temporal_closure(doc.tlinks) +For the temporal closure to be efficiently performed, on the back-end, the closure operation is executed with a point- +based reasoner which was inspired by the work of Gerevini et al. Gerevini et al. [1993]. As stated above, each TLink +instance contains an attribute named relation which is an instance of the TemporalRelation object. Within the +TemporalRelation all temporal relations are represented as the point relations by the means of a PointRelation +instance. In the point representation there are only four types of temporal relations, namely before (<), after (>), equal +(=), and not defined (None). With this point relation one can build a directed graph (henceforth referred as timegraph) +where the nodes are the entities endpoints (start and end of the entity) and the edges represent the before (<) relation. +This is accomplished by reflecting the after (>) relations and aggregating the equal (=) relations in a single node. +In the timegraph, inferring temporal relations is reduced to the problem of finding if two entities endpoints are connected, +i.e., they are in the same subgraph (by subgraph we mean a fully connected graph of the timegraph). If that is the case, +one can retrieve the endpoints on the entity pair and validate if the order of the entity endpoints is a valid temporal +relation. To clarify this concept, Figure 5 presents the timegraph built for a scenario where two tlinks were provided: +X MEETS Y and Y STARTS Z. To infer the temporal relation between X and Z one must query the endpoints in the +timegraph. In this case, one would get the following sequence of endpoints: sX < ex = sZ < eZ. After retrieving the +sequence of endpoints one just needs to validate if that sequence is a valid interval relation. In this example, one can +conclude that the temporal relation between X and Z is MEETS. +Figure 5: On the top part of the image is the relative relations between entities X, Y, and Z. On the bottom is the +graphical representation of the timegraph that would be generated. +To get a practical understanding of the runtime of the temporal closure algorithm, we executed it on all documents +currently available intieval. On a computer with an Intel Core i5-8500 CPU, the algorithm took less than half a second +for 95% of the documents, while the worst-case scenario took roughly 1.6 seconds. +This finalizes the presentation of the main functionalities, and some inner workings, of the first version of tieval. The +current version already provides functionalities that (we believe) will be beneficial for the TIE community. However, +we already have some ideas to further improve this library. These ideas are discussed in section 5. +4 +Observations +While building tieval, and in particular the datasets module, we found some inconsistencies in the corpus we were +working with. For instances, we found that the articles APW20000115.0209 and APW20000107.0088 of the AQUAINT +corpus contained the same news article, differing only in the annotations and in the value of the document creation +time. This type of inconsistencies were mitigated by implementing data cleaning processes that changed the original +annotations. Consequently, the results on the tieval framework will (most frequently) not resemble the exact result +that was reported in previous works, even if the same model is employed. +9 + +Relative Relation +Timegraphtieval +5 +Conclusion and Future Work +This work presented the first public release of the tieval package, an open-source Python library for the development +and evaluation of TIE systems. tieval provides several functionalities to facilitate research in this field. These include +the import of multiple benchmark corpora in different formats, domain-specific operations such as temporal closure or +transformation from interval to point relations, out-of-the-box baseline systems, and evaluation measures for TIE tasks. +Therefore, it provides the community with a standard way to benchmark TIE systems in a fair and comparable way, +while enabling the development of reproducible systems. +For future versions of the package, we aim to extend its functionalities. One idea we are keen to implement is +visualization techniques to display the relative timeline of events from the annotations. In addition, we will add methods +to include other levels of information when available such as coreference resolution in the MeanTime corpus Minard +et al. [2016] and causality relations in the TCR corpus Ning et al. [2018b]. We also intend to extend the list of supported +corpora and baseline models, in particular, to support corpora that cast the TIE task as a question-answer problem, such +as MCTaco Zhou et al. [2019] and TORQUE Ning et al. [2020]. This will allow us to produce a reproducibility study to +investigate several state-of-the-art systems and benchmark them in the different corpora. +References +Ricardo Campos, Gaël Dias, Alípio M Jorge, and Adam Jatowt. Survey of temporal information retrieval and related +applications. ACM Computing Surveys (CSUR), 47(2):1–41, 2014. +Artuur Leeuwenberg and Marie-Francine Moens. A survey on temporal reasoning for temporal information extraction +from text. Journal of Artificial Intelligence Research, 66:341–380, 2019. +Aakanksha Naik, Luke Breitfeller, and Carolyn Rose. Tddiscourse: A dataset for discourse-level temporal ordering of +events. 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Torque: A reading comprehension +dataset of temporal ordering questions. arXiv preprint arXiv:2005.00242, 2020. +11 + diff --git a/6NE3T4oBgHgl3EQfpgqP/content/tmp_files/load_file.txt b/6NE3T4oBgHgl3EQfpgqP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..64dc792e911626a1e9e49fb1bb8b18cd9c2c3226 --- /dev/null +++ b/6NE3T4oBgHgl3EQfpgqP/content/tmp_files/load_file.txt @@ -0,0 +1,564 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf,len=563 +page_content='tieval: AN EVALUATION FRAMEWORK FOR TEMPORAL INFORMATION EXTRACTION SYSTEMS Hugo Sousa 1,2, Alípio Jorge 1,2, and Ricardo Campos 1,3,4 1INESC TEC, Portugal 2University of Porto, Portugal 3Polytechnic Institute of Tomar, Portugal 4Ci2 - Smart Cities Research Center, Portugal {hugo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='sousa, alipio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='jorge, ricardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='campos}@inesctec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='pt January 12, 2023 ABSTRACT Temporal information extraction (TIE) has attracted a great deal of interest over the last two decades, leading to the development of a significant number of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Despite its benefits, having access to a large volume of corpora makes it difficult when it comes to benchmark TIE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' On the one hand, different datasets have different annotation schemes, thus hindering the comparison between competitors across different corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' On the other hand, the fact that each corpus is commonly disseminated in a different format requires a considerable engineering effort for a researcher/prac- titioner to develop parsers for all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This constraint forces researchers to select a limited amount of datasets to evaluate their systems which consequently limits the comparability of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Yet another obstacle that hinders the comparability of the TIE systems is the evaluation metric employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' While most research works adopt traditional metrics such as precision, recall, and F1, a few others prefer temporal awareness – a metric tailored to be more comprehensive on the evaluation of temporal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Although the reason for the absence of temporal awareness in the evaluation of most systems is not clear, one of the factors that certainly weights this decision is the necessity to implement the temporal closure algorithm in order to compute temporal awareness, which is not straightforward to implement neither is currently easily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' All in all, these problems have limited the fair comparison between approaches and consequently, the development of temporal extraction systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' To mitigate these problems, we have developed tieval, a Python library that provides a concise interface for importing different corpora and is equipped with domain-specific operations that facilitate system evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In this paper, we present the first public release of tieval and highlight its most relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The library is available as open source, under MIT License, at PyPI1 and GitHub2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Figure 1: tieval logo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 1https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='org/project/tieval/ 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='com/LIAAD/tieval arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='04643v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='CL] 11 Jan 2023 ti Valtieval 1 Introduction Understanding the temporal order of events is essential to human communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' We, humans, can easily understand the relative order of events in a conversation or when reading a news article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' However, many challenges are raised when we try to automate such tasks with a computer program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The first difficulty that emerges is how to represent temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Since in most cases we do not explicitly specify the start and end time of each event, temporal information, such as order and time span, ends up being inferred from the events themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' To this regard, computer algorithms can make use of temporal clues in the text, and of external sources, such as knowledge-bases, to anchor events on a timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' For instance, in the sentence “We went to dinner after the game.”, two events, “dinner” and “game”, can be automatically identified and used, despite the lack of explicit temporal information, to recreate a timeline of events (see Figure 2) supported on the word “after”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The ordering of events and the knowledge about them, can be further expanded if used together with appropriate external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' For instance, the event “game” can be contextualized and anchored on the timeline by searching for information on a knowledge-base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' However, in the case of the “dinner” event, it turns out impossible to know the exact time of occurrence unless it is specified in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This shows that representing temporal information is not a trivial task, since there are several borderline cases for which no standard approach has been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Figure 2: Relative timeline of events that can be inferred from the running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Over the years, and particularly in the last two decades, this problem has been highly studied, leading to several proposals from the research community Campos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2014], Leeuwenberg and Moens [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Most of the proposals were in the origin of the emergence of different annotation schemes and the various corpora that we have today at our disposal Naik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2019], Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2018a], UzZaman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Although these efforts have been essential to mature temporal information extraction and its subtasks – such as temporal expression identification or temporal relation classification – they also pose some problems upon the process of benchmarking different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' One of the problems has its roots in the fact that evaluating the methods, often requires reading multiple corpora, each of which has a different perspective on temporal representation, ultimately preventing comparability among the different methods and corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This is compounded by the fact that corpora are stored in a variety of formats (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=', XML, TimeML, or table ), which requires a considerable engineering effort to load them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Another issue that limits the comparison between systems is the lack of standardization in the metrics used in the evaluation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This is a particular problem of temporal relation extraction – a subtask of TIE, which deals with the identification and classification of the temporal relations between entities – where different metrics are often employed during the evaluation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' While initially systems were evaluated and compared using standard metrics, such as recall, precision, and F-score Verhagen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2007, 2010], more recently, metrics such as temporal awareness UzZaman and Allen [2011] have proven to be more reliable in the evaluation of temporal relation extraction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The reasoning behind this is that, while traditional metrics focus on the local effectiveness of the model, temporal awareness better understands the relative order of events by considering the global temporal structure of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This is accomplished by taking into account the temporal relations that can be inferred from the established ones (a process typically referred to as temporal closure), making this a more comprehensive metric for evaluating temporal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Despite the emergence of this temporal awareness, many studies still rely solely on traditional metrics to evaluate their system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' We speculate that this is due to the fact that temporal awareness requires domain-specific operations such as temporal closure – which are not (yet) readily available in every framework and therefore require individual implementation by each research group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In addition, temporal awareness requires the implementation of a strategy to deal with inconsistent predictions of the system, which is generally not explored in recent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' To mitigate the above issues, we developed tieval, a Python library that enables the development and evaluation of TIE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This framework provides a simple interface to download and read TIE corpora in various formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' It currently covers well-known corpus – such as TempEval-3 UzZaman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2013], TDDiscource Naik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2019], and MeanTime Minard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2016] – however, it lays the foundations for others to be included by providing base classes for the construction of the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' It also provides domain-specific operations – such as temporal closure and simple translation of intervals into point relations – that can be used to develop TIE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In addition to this, it includes an evaluation infrastructure for a comprehensive assessment of the effectiveness of the different models being evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Because tieval supports the entire development pipeline of TIE, it can also be used to ensure reproducibility and fair benchmarking of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The main contributions of tieval are the following: 2 tieval 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' it gathers the multiple corpora for the development of TIE systems, making it easy to access with just a few lines of code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' it facilitates access to domain-specific operations, such as interval to point relation and temporal closure, as well as metrics such as temporal awareness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' it provides a standard framework, thus making it easy for new methods to be compared against previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The remaining of the paper is organized as follows: The next section, provides an overview of recent work in TIE and some of its software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' We then proceed to present the tieval package in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' We start with a general introduction and then go into some of its most relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Section 5 serves to present our thoughts on what we strive to be next steps in the development of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 2 Related Work Extracting temporal information from documents written in natural language in an inter-operable format has long been an interest of the artificial intelligence community Ling and Weld [2010], Derczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Since the introduction of the Time Markup Language (TimeML) Pustejovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2003a], in 2003, the temporal graph has become the de-facto standard to represent temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In this graph, the nodes are temporal entities and the edges are the temporal relation that hold between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The temporal entities can take two forms: event expressions, which are defined as situations that happened (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=', “went” or “bought”);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' and temporal expressions (timex), which can convey temporal information explicitly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=', “October 27, 199”) or implicitly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=', “a few years ago”) Campos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The temporal relations are held in the form of temporal links (tlink) that contain temporal relations between pairs of events (E-E relations), events and time expressions (E-T relations), and events and document creation time (E-DCT relations), where DCT is a special timex that stores document creation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Overall, these temporal relations can take thirteen types, which is the number of relations that can exist between two time intervals Allen [1983].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The first corpus that was annotated with this scheme was TimeBank Pustejovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2003b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The release of this corpus, dated from 2003, sparked a wave of research in the field later on also used on the TempEval shared tasks UzZaman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2013], Verhagen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2007, 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' These tasks end up segmenting TIE into a set of sub-problems that can be conceptually defined as temporal entity identification, tlink identification, and tlink classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Although some works developed systems for more than one of these sub-tasks, most of the systems are concerned with only one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Furthermore, temporal entity identification systems are traditionally partitioned into subsystems for the several classes of temporal entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' For example, for the TimeBank corpus, one system is usually trained to identify events and another to identify timexs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The tieval architecture follows this natural decomposition of the TIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The TimeBank corpus, and more abstractly, the TimeML annotation scheme was widely studied by the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Such scrutiny lead to the emergence of several new corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Some used the TimeML annotation scheme to create new corpora, such as AQUAINT Graff [2002] and the Platinum corpus UzZaman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2013], while others were concerned in extending the annotation scheme to other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The most remarkable effort on this domain was the TempEval-2 shared task Verhagen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2010] that produced corpora for Chinese Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2014], French Bittar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2011], Italian Caselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2011], and the Spanish Nieto and Saurí [2012] language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Another noteworthy effort is the MeanTime corpus Minard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2016] in which the authors annotated 120 news articles written in English from Wikinews3, and translated the texts into Italian, Spanish, and Dutch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Costa and Branco Costa and Branco [2012] followed a similar process to construct TimeBankPT, translating the original TimeBank to Portuguese and adapting the annotations when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Apart from the extensions to other languages, the TimeML annotation scheme was also extended to other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' A concrete example is the case of the clinical domain for which two corpora have been produced, the i2b2 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2013] and THYME Styler IV et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2014]4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Further significant contributions were the proposals that explored ways to mitigate some of the issues found on the TimeBank annotation effort, such as: sparse annotation – TimeBank-Dense Cassidy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2014] and TDDiscourse Naik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2019];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' improve inter-annotator agreement – MATRES Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2018a];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' and include other sources of knowledge – TCR Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2018b] and RED O’Gorman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Aside from the TimeML, and related approaches, there have also been other proposals that were explored by the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' One of them is absolute timeline placement, in which the temporal entities are directly anchored on a timeline by labeling each entity with the time (or time span) of occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The most remarkable efforts in this direction were produced by Reimers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Reimers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2016] – which produced the EventTime corpus by annotating the events in TimeBank with a specific day, or span of days – and Leeuwenberg and Moens Leeuwenberg and Moens 3https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='wikinews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='org/ 4These corpora are not available for open access and, as a consequence, we were not able to include them on the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 3 tieval [2020] – which annotated 169 clinical records from the i2b2 corpus with the most likely start and end time of each event along with a lower and upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This shows that several corpora have been introduced for the TIE task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' However, the fact that they were released in different formats makes it hard to leverage their power, which is one of the issues mitigated by tieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' To the best of our knowledge, the only framework that made available TIE operations – including temporal closure and temporal awareness – is the Anafora Tools project5 which was built to work with files stored in the Anafora XML format Chen and Styler [2013], used to annotate the THYME corpus Styler IV et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The framework presented in this paper aims to be a more general tool, unifying all corpora in a single format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 3 tieval Our vision for tieval was to build a framework that would support and facilitate the evaluation of TIE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' With the development of libraries such as Numpy, TensorFlow, and PyTorch, Python has established itself as the programming language of choice within the machine learning community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' For that reason, we built tieval in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' To facilitate the installation we made it available on Python Package Index (PyPI)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Thus, the toolkit can be easily installed through pip, as follows: $ pip install tieval==0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='6 In this paper, we will use version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='6, which is the first and the most recent version of the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' However, the reader is advised to install the newest release at the time of reading the paper and refer to the project repository for up-to-date documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Furthermore, for users that might be interested in contributing to tieval, we encourage forking the source repository and making a pull request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' tieval contains three modules that represent the three cornerstones of any machine learning project: datasets, models, and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The datasets module is responsible for downloading and reading the corpora available for TIE, the models module is responsible for the construction of the models, and the evaluation module has methods to make a proper evaluation for each of the TIE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In the following sections, we will present the inner workings of the framework with scripts to exemplify the usability of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='1 Datasets With tieval, we wanted to mitigate the issues referred above by making it easy for the user to work with several corpora with a few lines of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' To that end, we developed an architecture that would unify the different annotations and storing formats of the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This architecture is composed of several objects which are depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Figure 3: Objects used to represent a dataset on tieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The arrow represent a relation of “Iterable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The Dataset object is the final representation of each corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' It compiles the set of all the documents in the corpus on the documents attribute which is segmented into the train and test attributes whenever provided in the original paper7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Each document is then stored as an instance of the Document class (see the Document grey box in Figure 3), which contains all the information necessary for TIE, more specifically: name a string that contains the name of the document (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' “wsj_0026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='tml”);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='com/bethard/anaforatools 6https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='org/project/tieval/ 7When no standard train/test split is provided by the authors all the documents are placed on the train attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 4 Dataset Document Entity TLink .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='documents .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='name .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='text .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='source .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='train .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='text .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='value .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='target .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='test .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='dct .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='endpoints .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='relation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='entities **kwargs **kwargs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='tlinkstieval text a string with the raw text of the document;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' dct is a Timex that contains the document creation time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Timex("12-10-2004"));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' entities is the set of Entities – either a Timex or Event – that are annotated on the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Each Entity is, at is core, a data class made to store all the info provided on the annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Therefore, it has to be flexible to accommodate for the different types of information provided in different corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' For instances, the GraphEve corpus provides the lemma for each event while TempEval-2 does not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' tlinks a set o TLink’s that stores the temporal relations annotated on the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Each TLink contains a source and target entity as well as the temporal relation between them – on the relation attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' A special remark needs to be made about the relation attribute of the TLink object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' When initiating a TLink instance one needs to pass the temporal relation that holds between the two temporal entities (the source and the target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In most of the corpora this is one of the thirteen temporal relations Allen [1983] that can hold between two time intervals, however, there are corpora where the annotators were more flexible on the type of relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Examples of this are the TempEval-2 and the MATRES corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' On TempEval-2 the annotators were allowed to give more ambiguous relations as “BEFORE-OR-OVERLAP” and “OVERLAP-OR-AFTER”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In MATRES the annotators were asked to provide the temporal relation between the start points of the temporal entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In order to accommodate the several types of annotations, we build TemporalRelation object, which handles the relation that was annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Inside this object, every relation is represented in point relations – instead of the traditional interval relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Figure 4 shows how to represent the interval relation “BEFORE” into a point relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' A relative relation is also included in the figure for illustrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Figure 4: Relative timeline of events that can be inferred from the running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Note that the “BEFORE-OR-OVERLAP” relation on TempEval-2 represents an uncertainty of the annotator between the end time of the source entity and the start time of the target entity, however, the annotator is certain about the remaining point relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Further note that, although we explicitly state four-point relations in Figure 4, upon the adaptation of the current datasets into tieval format, three of them are redundant, as the point relation “end A < start B” completely defines the remaining point relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Therefore, on tieval, whenever there is a new dataset to include, the user can provide the relation in the way that is most appropriate, as shown in Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Listing 1: Different ways to pass the temporal relation to the TLink object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The first argument (X) is the source entity, the second (Y) is the target entity, and the third is the temporal relation between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This can be passed as an interval relation, “before”, or as a point relation, in the form of a dictionary structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' On the latter, the interpretation for the expected keys is the following: “x” and “y” stands for the source and target entity, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' while the “s” and “e” stand for “start” and “end”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' As an example, “xe_ys” is the point relation between the source end and the target start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' from tieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='links import TLink tl1 = TLink("X", "Y", "before") tl2 = TLink("X", "Y", {"xe_ys": "<"}) tl3 = TLink("X", "Y", {"xs_ys": "<", "xs_ye": "<", "xe_ys": "<", "xe_ye": "<" ,}) In order to reach a standardized representation for the different corpora, we developed a reader for each of the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Each dataset reader has inherited from an abstract base class, named BaseDocumentReader, which requires the implementation of five methods named after the five attributes used to create an instance of a Document: name, text, dct, entities, and tlinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' To extract this information, the base class contains three attributes: the path for the document being read;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' the content of the dictionary produced by parsing the document with the xmltodict8 library;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' and the xml attribute that results from parsing the file with the xml9 library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Note that, while json is nowadays the standard format for the 8https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='org/project/xmltodict/ 9https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='org/3/library/xml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='etree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='elementtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='html 5 Relative Relation Interval Relation Point Relationtieval exchange of the information, we had to resort to xml as most datasets were stored in that format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The script presented in Listing 2 illustrates how to read a document from the TempEval-3 corpus with the TempEval3DocumentReader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Listing 2: Read a document of the TempEval-3 corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' from tieval import datasets path = "tempeval -3/ wsj_0026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='tml" reader = datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='TempEval3DocumentReader(path) doc = reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='read() To fully integrate a new corpus on the library – and automatically read the entire corpus – the user just needs to add an entry on the DATASETS_METADATA dictionary with the metadata necessary to read the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This information will be used on the read function of the datasets module, which only requires the name of the corpus to produce an instance of the Dataset object with all the annotations provided in there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The script in Listing 3 presents how to perform such operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Listing 3: Read the TempEval-3 corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' from tieval import datasets te3 = datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='read("TempEval -3") The current version of tieval natively supports the download and reading of an extensive list of corpora for TIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' A complete list of the corpora considered is provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In order to ensure long-term support for these corpora, we created a repository with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Besides that, it also has the advantage that we can standardize the structure of the folders and add useful information to the raw datasets (for instance, the spans of the temporal entities identified on the text) and fix errors on the original annotation10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' For that reason, we were careful to verify the license for each of the corpora and publish only the ones that allowed for redistribution or did not provide any license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Table 1: The corpora currently supported on tieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Language # Docs # Events # Timexs # Tlinks AncientTimes Arabic 5 0 106 0 Dutch 5 0 130 0 English 5 0 311 0 French 5 0 290 0 German 5 0 196 0 Italian 5 0 234 0 Spanish 5 0 217 0 Vietnamese 4 0 120 0 Aquaint English 72 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='351 639 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='832 EventTime English 36 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='498 0 0 GraphEVE English 103 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='298 0 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='204 KRAUTS German 192 0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='282 0 MATRES English 274 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='065 0 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='504 MeanTime English 120 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='882 349 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='753 Spanish 120 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='000 344 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='975 Dutch 120 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='346 346 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='487 Italian 120 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='980 338 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='675 Narrative Container English 63 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='559 439 737 Continued on next page 10The changes made on the original corpus are detailed on the file logbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='rst in the docs folder of the project repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 6 tieval Table 1: The corpora currently supported on tieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' (Continued) Professor Heideltime English 24,642 0 254,803 0 French 27,154 0 83,431 0 German 19,095 0 194,043 0 Italian 9,619 0 58,823 0 Portuguese 24,293 0 111,810 0 Spanish 33,266 0 348,011 0 Platinum (TempEval-3) English 20 748 158 929 TimeBank Spanish 210 12,384 1,532 21,107 French 108 2,115 533 2,303 Portuguese 182 7,887 1,409 6,538 English 183 6,681 1,426 5,120 TimeBank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='2 English 183 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='940 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='414 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='413 TCR English 25 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='134 242 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='515 TDDiscourse English 34 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='101 0 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='150 TempEval 2 Chinese 52 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='783 946 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='802 English 182 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='656 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='390 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='945 French 83 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='301 367 372 Italian 64 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='377 653 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='884 Korean 18 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='583 317 0 Spanish 210 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='384 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='502 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='304 English 275 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='780 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='223 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='881 TimeBank Dense English 36 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='712 289 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='715 TrainT3 (TempEval-3) Spanish 175 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='686 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='269 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='283 Wikiwars English 22 0 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='662 0 German 22 0 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='239 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='2 Models The current version of tieval has four built-in models, namely: a baseline for timex identification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' the HeidelTime model Strötgen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2013] for timex identification and classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' a baseline for event identification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' and the CogCompTime 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='0 model Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2019] for tlink classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The availability of these four models is intended for practitioners that may want to experiment using any layer of temporal information in their specific application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Apart from that, it also provide researchers the implementation of baseline models for reference in their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' For the baseline models, we provide pre-trained weights, however, the user can also train the model from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' A description of each of the models is provided below: TimexIdentificationBaseline For this baseline we trained – from scratch – the spaCy11 named entity recognition model to identify the timexs on the TempEval-3 corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' EventIdentificationBaseline This model has the same architecture of the TimexIdentificationBaseline but was trained to identify events rather than timexs on the TempEval-3 corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' HeidelTime This model is a widely recognized multilingual temporal tagger which was original written in Java12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' However there have been efforts to build python wrappers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In tieval we used the py_heideltime wrapper which is available on GitHub13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 11https://spacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='io/ 12https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='com/HeidelTime/heideltime 13https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='com/hmosousa/py_heideltime 7 tieval CogCompTime2 This model leverages the ELMo Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2018] word embeddings and the TempProb Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2018c] knowledge base to classify the temporal relation between a pair of temporal entities Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Our implementation was adapted from the repository made available14 by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Listing 4 presents a script that would download the baseline model for temporal expression identification (TimexIdentificationBaseline), train the model on the TempEval-3 train set, and produce predictions for the TempEval-3 test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Listing 4: How to download, train, and predict with for the temporal identification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' from tieval import models model = model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='TimexIdentificationBaseline () model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='fit(te3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='train) predictions = model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='predict(te3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='test) A user interested in releasing his/her model in tieval can do it by creating a subclass of one of our base classes for models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' There are two base classes: a BaseModel which just requires the implementation of the predict method which is intended for models that are available in other repositories – for instance, the HeidelTime model – and a BaseTrainableModel which, besides the predict, requires the implementation of the fit method, which implements the training loop for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='3 Evaluation tieval provides an evaluation function for four subtasks of TIE, more specifically: timex identification, event identification, tlink identification, and tlink classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Table 2: The results obtained by evaluating the four models integrated in tieval on the Platinum (TempEval-3 test set), TCR, and MeanTime (the English version) corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' P stands for precision, R for recall, F1 is the F1-score, and TF1 is the temporal awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' All the results in the table are micro metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Platinum TCR MeanTime P R F1 (TF1) P R F1 (TF1) P R F1 (TF1) TimexBaseline 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='5 HeidelTime 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='8 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='8 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='8 EventBaseline 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='6 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='8 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='1 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='9 CogCompTime2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='7 (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='3) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='4 (69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='3) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='6 (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='9) The input is standard for all the evaluation functions: annotations, a dictionary with the name of the documents as keys and the annotations as values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' predictions, follows the same structure of the annotations but for each document key contains the predictions made by a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The output of the functions is dependent on the task being evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' For the identification tasks (timex, event, and tlink) the function produces the standard macro and micro metrics for precision, recall, and f1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Listing 5 presents a script that evaluates the predictions made by the event baseline model in the TempEval-3 test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Listing 5: Evaluate event baseline model on the TempEval-3 test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' from tieval import evaluate annotations = {doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='name: doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='events for doc in te3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='test} result = evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='event_identification(annotations , predictions) Table 2 depicts the results obtained by the implemented on benchmark corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Note that TF1 is the temproal awareness metric and is only computed for CogCompTime2 (the only tlink classification system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Another interesting remark is the fact that the TimexBaseline achieves effectiveness comparable to HeidelTime despite its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The tlink classification is the most elaborate evaluator as it also computes the temporal awareness metric UzZaman and Allen [2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The complexity of the calculation of temporal awareness lies in the computation of temporal closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' With temporal_closure the closure operation can be easily performed on the document level, with the closure method of 14https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='com/qiangning/NeuralTemporalRelation-EMNLP19 8 tieval the Document object, or applied to a set of tlink’s with the temporal_closure function available on the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The script in Listing 6 illustrates how to perform such operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Listing 6: How to compute the temporal closure with a Document object and with a set of TLink’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' from tieval import temporal_closure doc = te3["wsj_0026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='tml"] closure_tlinks = doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='closure closure_tlinks = temporal_closure(doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='tlinks) For the temporal closure to be efficiently performed, on the back-end, the closure operation is executed with a point- based reasoner which was inspired by the work of Gerevini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Gerevini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [1993].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' As stated above, each TLink instance contains an attribute named relation which is an instance of the TemporalRelation object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Within the TemporalRelation all temporal relations are represented as the point relations by the means of a PointRelation instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In the point representation there are only four types of temporal relations, namely before (<), after (>), equal (=), and not defined (None).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' With this point relation one can build a directed graph (henceforth referred as timegraph) where the nodes are the entities endpoints (start and end of the entity) and the edges represent the before (<) relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This is accomplished by reflecting the after (>) relations and aggregating the equal (=) relations in a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In the timegraph, inferring temporal relations is reduced to the problem of finding if two entities endpoints are connected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=', they are in the same subgraph (by subgraph we mean a fully connected graph of the timegraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' If that is the case, one can retrieve the endpoints on the entity pair and validate if the order of the entity endpoints is a valid temporal relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' To clarify this concept, Figure 5 presents the timegraph built for a scenario where two tlinks were provided: X MEETS Y and Y STARTS Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' To infer the temporal relation between X and Z one must query the endpoints in the timegraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In this case, one would get the following sequence of endpoints: sX < ex = sZ < eZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' After retrieving the sequence of endpoints one just needs to validate if that sequence is a valid interval relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In this example, one can conclude that the temporal relation between X and Z is MEETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Figure 5: On the top part of the image is the relative relations between entities X, Y, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' On the bottom is the graphical representation of the timegraph that would be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' To get a practical understanding of the runtime of the temporal closure algorithm, we executed it on all documents currently available intieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' On a computer with an Intel Core i5-8500 CPU, the algorithm took less than half a second for 95% of the documents, while the worst-case scenario took roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='6 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This finalizes the presentation of the main functionalities, and some inner workings, of the first version of tieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' The current version already provides functionalities that (we believe) will be beneficial for the TIE community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' However, we already have some ideas to further improve this library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' These ideas are discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 4 Observations While building tieval, and in particular the datasets module, we found some inconsistencies in the corpus we were working with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' For instances, we found that the articles APW20000115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='0209 and APW20000107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content='0088 of the AQUAINT corpus contained the same news article, differing only in the annotations and in the value of the document creation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This type of inconsistencies were mitigated by implementing data cleaning processes that changed the original annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Consequently, the results on the tieval framework will (most frequently) not resemble the exact result that was reported in previous works, even if the same model is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' 9 Relative Relation Timegraphtieval 5 Conclusion and Future Work This work presented the first public release of the tieval package, an open-source Python library for the development and evaluation of TIE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' tieval provides several functionalities to facilitate research in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' These include the import of multiple benchmark corpora in different formats, domain-specific operations such as temporal closure or transformation from interval to point relations, out-of-the-box baseline systems, and evaluation measures for TIE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Therefore, it provides the community with a standard way to benchmark TIE systems in a fair and comparable way, while enabling the development of reproducible systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' For future versions of the package, we aim to extend its functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' One idea we are keen to implement is visualization techniques to display the relative timeline of events from the annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' In addition, we will add methods to include other levels of information when available such as coreference resolution in the MeanTime corpus Minard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2016] and causality relations in the TCR corpus Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2018b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' We also intend to extend the list of supported corpora and baseline models, in particular, to support corpora that cast the TIE task as a question-answer problem, such as MCTaco Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2019] and TORQUE Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' This will allow us to produce a reproducibility study to investigate several state-of-the-art systems and benchmark them in the different corpora.' metadata={'source': 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temporal expressions in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' New Directions in Question Answering, 3:28–34, 2003a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Ricardo Campos, Gaël Dias, Alípio Mário Jorge, and Célia Nunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Identifying top relevant dates for implicit time sensitive queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' Information Retrieval Journal, 20(4):363–398, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE3T4oBgHgl3EQfpgqP/content/2301.04643v1.pdf'} +page_content=' James F Allen.' metadata={'source': 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a/8NAzT4oBgHgl3EQfE_qO/content/tmp_files/2301.01003v1.pdf.txt b/8NAzT4oBgHgl3EQfE_qO/content/tmp_files/2301.01003v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..262c34dc61e446a3e6b4bcf14b151c67cc5d65d2 --- /dev/null +++ b/8NAzT4oBgHgl3EQfE_qO/content/tmp_files/2301.01003v1.pdf.txt @@ -0,0 +1,867 @@ +Astronomy & Astrophysics manuscript no. main +©ESO 2023 +January 4, 2023 +Letter to the Editor +Polarised radio pulsations from a new T dwarf binary +H. K. Vedantham1, 2, Trent J. Dupuy3, E. L. Evans3, A. Sanghi4, J. R. Callingham1, 5, T. W. Shimwell1, 5, W. M. J. +Best5, M. C. Liu6 and P. Zarka7 +1 ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, Dwingeloo, 7991 PD, The Netherlands +2 Kapteyn Astronomical Institute, University of Groningen, PO Box 72, 97200 AB, Groningen, The Netherlands +3 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK +4 The University of Texas at Austin, Department of Astronomy, 2515 Speedway, C1400, Austin, TX 78712, USA +5 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA, Leiden, The Netherlands +6 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA +7 LESIA, CNRS – Observatoire de Paris, PSL 92190, Meudon, France +Received XXX; accepted YYY +ABSTRACT +Brown dwarfs display Jupiter-like auroral phenomena such as magnetospheric Hα emission and coherent radio emission. Coherent +radio emission is a probe of magnetospheric acceleration mechanisms and provides a direct measurement of the magnetic field strength +at the emitter’s location, both of which are difficult to access by other means. Observations of the coldest brown dwarfs (spectral +types T and Y) are particularly interesting as their magnetospheric phenomena may be very similar to those in gas-giant exoplanets. +Here we present 144 MHz radio and infrared adaptive optics observations of the brown dwarf WISEP J101905.63+652954.2 made +using the LOFAR and Keck telescopes respectively. The radio data shows pulsed highly circularly polarised emission which yields a +rotation rate of 0.32 ± 0.03 hr−1. The infrared imaging reveals the source to be a binary with a projected separation of 423.0 ± 1.6 mas +between components of spectral type T5.5 ± 0.5 and T7.0 ± 0.5. With a simple “toy model” we show that the radio emission can +in principle be powered by the interaction between the two dwarfs with a mass-loss rates of at least 25 times the Jovian value. +WISEP J101905.63+652954.2 is interesting because it is the first pulsed methane dwarf detected in a low radio-frequency search. +Unlike previous gigahertz-frequency searches that were only sensitive to objects with kiloGauss fields, our low-frequency search is +sensitive to surface magnetic fields of ≈ 50 Gauss and above which might reveal the coldest radio-loud objects down to planetary +mass-scales. +1. Introduction +High energy charges around brown dwarfs are expected to be +created by auroral (or magnetospheric) processes akin to that +seen on gas-giant planets, as opposed to coronal and chro- +mospheric acceleration expected on stars (Nichols et al. 2012; +Williams 2018; Turnpenney et al. 2017). This paradigm has +been established based on highly circularly polarised and ro- +tationally modulated radio pulses and Hα emission observed +on brown dwarfs (Hallinan et al. 2007, 2008, 2015; Route & +Wolszczan 2012, 2016a; Williams et al. 2017). The radio emis- +sion is of particular interest because it is expected to occur at +the local cyclotron frequency, which in the non-relativistic limit, +only depends on the ambient magnetic field strength (Melrose +& Dulk 1982). Because Zeeman splitting observations become +very challenging in such cold objects as brown dwarfs due to +the lack of non-broadened spectral lines, radio observations may +be the only viable technique to directly measure their magnetic +field strengths and topologies. In addition, unlike rocky plan- +ets, gas giants and brown dwarfs have predictable and relatively +simple internal structures at depths where their magnetic field +is expected to be generated (Chabrier & Baraffe 2000). This +makes them ideal targets to test dynamo scaling laws (e.g., Chris- +tensen et al. 2009) that are likely applicable even in the exoplanet +regime. +Despite concerted searches, radio detections of the cold- +est brown dwarfs are rare. The coldest, spectral type Y brown +dwarfs have not been detected in the radio (Kao et al. 2019). At +the warmer spectral type T, four brown dwarfs have been de- +tected in dedicated surveys at radio frequencies of 5 GHz and +above (Route & Wolszczan 2012; Kao et al. 2016; Route & +Wolszczan 2016b,a; Kao et al. 2016, 2018). Recently, we re- +ported the first direct discovery of a brown dwarf made by virtue +of its radio emission (Vedantham et al. 2020) using the LO- +FAR radio telescope (van Haarlem et al. 2013) at 144 MHz. +Here we report our second discovery also made with LO- +FAR. WISEP J101905.63+652954.2 was originally discovered +by Kirkpatrick et al. (2011) in Wide-field Infrared Survey Ex- +plorer (WISE) data (Wright et al. 2010) and, using spectroscopic +data, assigned optical and near-infrared spectral types of T7 and +T6, respectively. +Cold brown dwarfs share their radio phenomenology with +Jupiter. The radio emission consists of two components. A quasi- +quiescent component that is unpolarised or weakly polarised and +a highly circularly polarised pulsed component that repeats at +the rotation rate (Williams 2018; Antonova et al. 2013; Hallinan +et al. 2008; Berger 2006). However, the radio energetics of the +detected brown dwarfs is orders of magnitude larger than that +seen in Jupiter. This combined with a lack of detection of UV or +H3+ from brown dwarfs (Saur et al. 2021; Gibbs & Fitzgerald +2022) suggest that the Jovian auroral energetics cannot be simply +scaled to brown dwarfs. In any case, magnetic field lower lim- +its derived from the pulsed component in three of the detected +T dwarfs have been over a factor of three larger than the pre- +dictions of leading dynamo scaling laws that can successfully +predict the field strength of some solar system planets and low +mass stars (Kao et al. 2018). This suggests that the model is +inadequate, or it is also possible that by virtue of observing at +Article number, page 1 of 7 +arXiv:2301.01003v1 [astro-ph.SR] 3 Jan 2023 + +A&A proofs: manuscript no. main +high frequencies, the previous radio surveys were only sensitive +to objects with anomalously large magnetic fields. For instance, +Christensen et al. (2009) predict a field strength of 103 Gauss for +a 50 MJup brown dwarf with an age of 109 yr and a surface tem- +perature of 1500 K (late-L dwarf). The corresponding peak cy- +clotron frequency in its magnetosphere is about 2.8 GHz which +will make such an object undetectable in a 5 GHz survey even +if it were ‘typical’ of the predicted population. The 144 MHz +LOFAR data can detect objects with surface field strengths as +low as 50 G. Therefore, the LOFAR-detected objects such as +WISEP J101905.63+652954.2 are beginning to provide a more +complete sample to critically test dynamo scaling laws over a +larger range in magnetic field strengths. +This paper is organised as follows: §2 presents details of the +radio and infrared observations and the analysis of the radio light +curve. In §3 we discuss the possible mechanisms driving the ra- +dio emission, and present our conclusions and outlook in §4. +2. Observations +2.1. LOFAR 144 MHz observations +WISEP J101905.63+652954.2 +was discovered as part of our +ongoing search (e.g. Callingham et al. 2021) for stars, brown +dwarfs, and exoplanets using data from the LOFAR Two Metre +Sky Survey (LoTSS; Shimwell et al. 2022). Our methodology +has typically involved searching for circularly polarised sources +in deep 8 hr exposure LoTSS images. This is how we found +Elegast, the first radio-selected brown dwarf (Vedantham et al. +2020). Because brown dwarf auroral emission is typically pulsed +at the rotation period, we have since implemented a search al- +gorithm to construct Stokes-V light curves on various time-bin +widths and search for on-off and periodic pulsations from known +brown dwarfs. Although we plan to conduct an untargeted search +for such pulses over the Northern sky, we first validated our ap- +proach by a targeted search of ten known T- and Y-type brown +dwarfs which led to the discovery of Stokes-V radio pulsations +from WISEP J101905.63+652954.2. +Our current pipeline takes in the standard calibrated visi- +bilities from the LoTSS survey. We first subtract the LoTSS- +detected sources from the visibilities using their direction depen- +dent gains while retaining on-axis sources in the direction of the +target for an additional round of self-calibration as described in +van Weeren et al. (2021). We then modelled and subtracted these +sources using their clean components from wsclean. We then +imaged the target fields using wsclean (Offringa et al. 2014) at +a cadence of 4 minutes and extracted the light curves from these +images after averaging over the available bandwidth. The light +curve of WISEP J101905.63+652954.2 shows a statistically sig- +nificant burst (Fig. 1). The on and flanking off-burst snapshot im- +ages are also shown. The figure also shows the light curve binned +to a resolution of 16 min in red that reveals a hint of periodicity +at around a 3 hr period. +Polarised radio emission from planets and brown dwarfs is +expected to have a periodic signature at the rotation period of +the object due to beaming (akin to a light house). To ascertain +the period signature in the light curve, we computed the Lomb- +Scargle periodogram of the light curve using the astropy (As- +tropy Collaboration et al. 2013, 2018) implementation (See Fig. +2). To compute the significance of the periodogram peaks we +computed the false alarm rate based on the bootstrap method de- +scribed in VanderPlas (2018). We detect a dominant peak at a +frequency of 0.324 hr−1 with a false alarm rate of under 3%. We +compute an uncertainty in the peak’s location of 0.033 hr−1 using +the method prescribed in equation 52 of VanderPlas (2018). +2.2. Keck/NIRC2 LGS AO +We observed WISEP J101905.63+652954.2 on 2015 January +15 UT and 2022 January 24 UT with the facility imager NIRC2 +at the Keck II telescope in concert with the laser guide star (LGS) +adaptive optics (AO) system (van Dam et al. 2006; Wizinowich +et al. 2006). For tip-tilt correction, we used the star USNO- +B1.0 1554-0140735, which is 23′′ away from the target and pro- +vided flux to the tip-tilt sensor equivalent to R = 18.2 mag. The +wavefront sensor monitoring the LGS measured flux equivalent +to a V = 10.2 mag star in 2015 and V = 8.5 mag in 2022, thanks +to the intervening LGS upgrade (Chin et al. 2016). We obtained +images with standard Maunakea Observatories filters in the J +and H bands (Tokunaga et al. 2002) as well as CH4s, a medium- +bandwidth filter centred on the H-band flux peak of T dwarfs. +For each filter, we obtained between four and six dithered 180-s +images in 2015 and 60-s images in 2022 while keeping the LGS +fixed to the centre of NIRC2’s narrow camera (0′′.01 pixel−1) +field-of-view (10′′ × 10′′). In 2015, the AO correction deterio- +rated significantly as we collected data, and the quality of our +H-band data set was too poor to be included in our analysis. +We reduced our data using the same custom scripts as in +our previous work (e.g., Liu et al. 2008; Dupuy et al. 2015), +and examples of individual exposures are shown in Figure 3. +We measured the separation, position angle (PA), and magnitude +difference in individual exposures using three-component, two- +dimensional Gaussians, and computed the means and standard +deviations of measurements from individual exposures as the fi- +nal measurements and their uncertainties. For our 2015 data, we +used the astrometric calibration of Yelda et al. (2010) to correct +for nonlinear distortion, the orientation of NIRC2 (by subtracting +0◦.252), and the pixel scale (9.952±0.002 mas pixel−1). Likewise, +for our 2022 data we used the calibration of Service et al. (2016). +The resulting binary parameters are given in Table 1. Our relative +astrometry is consistent within the errors at each epoch, and the +repeated observations in J and CH4s filters show no significant +change in flux ratio. +To compute the final relative astrometry at each epoch, we +took the weighted average of our relative astrometry measure- +ments and added a systematic error of 1.5 mas to account for the +uncertainty in the distortion corrections of NIRC2. This gives +separations of 423.0 ± 1.6 mas and 468.2 ± 1.6 mas and PAs of +161◦.71±0◦.23 and 166◦.87±0◦.20, in 2015 and 2022, respectively. +The observed motion of ≈ 7 mas yr−1 is much lower than the +proper motion of the system (150.6 ± 1.1 mas yr−1) measured by +Kirkpatrick et al. (2019), so we conclude the companion shares +a common proper motion with the primary and is physically +bound. +Our Keck images also showed a fainter point source ≈2′′ +away from WISEP J101905.63+652954.2 at a position angle of +≈200◦. We identified this source in the Pan-STARRS1 3π Survey +catalog (Chambers et al. 2016) as PSO J154.7727+65.4978. It is +visible in stacked rizyP1 images and appears brightest in zP1. Its +(z − y)P1 = 0.41 ± 0.13 mag color (using stacked photometry) is +far too blue to be a fainter, later-T or Y dwarf (Best et al. 2018), +so we conclude this is a background star or galaxy. Although this +source is only about 2′′ from the nominal position of the radio +detection, it is almost certainly not the source of the observed +radio emission. The high circular polarisation in the radio-band +is inconsistent with an extragalactic origin, so we only need con- +sider the Galactic stellar hypothesis. The absolute radio astrom- +Article number, page 2 of 7 + +Vedantham et al.: Radio pulsation from new T-dwarf binary +10h19m15s +10s +05s +00s +65±3003000 +0000 +2903000 +0000 +RA (J2000) +Dec (J2000) +Stokes V +10h19m15s +10s +05s +00s +65±3003000 +0000 +2903000 +0000 +RA (J2000) +Dec (J2000) +Stokes V +10h19m15s +10s +05s +00s +65±3003000 +0000 +2903000 +0000 +RA (J2000) +Dec (J2000) +Stokes V +(a) +(b) +(c) +(d) +Fig. 1. Panel (a) shows the Stokes-V radio lightcurve of WISEP J101905.63+652954.2 with a bin width of 4 minutes (black points with ±1σ +errorbars) and 16 minutes (red curve with shaded ±1σ uncertainty). The point marked with the black square is a significant detection with a +flux-density of −4.1(7) mJy whose Stokes-V image is shown in panel (c). Panels (b) and (d) show similar 4 min exposure images bracketing the +integration show in panel (c). +Table 1. Keck LGS AO Relative Astrometry and Photometry of WISEP J101905.63+652954.2. +Epoch (MJD) +Filter +Separation (mas) +PA (deg) +∆m (mag) +57037.5382 +J +416 ± 7 +161.5 ± 0.6 +0.37 ± 0.06 +57037.5246 +CH4s +423.1 ± 0.6 +161.72 ± 0.12 +0.489 ± 0.019 +59603.5270 +J +467.2 ± 1.1 +166.78 ± 0.12 +0.494 ± 0.021 +59603.5218 +CH4s +467 ± 3 +166.82 ± 0.18 +0.48 ± 0.03 +59603.5174 +H +468.7 ± 0.7 +166.97 ± 0.10 +0.579 ± 0.013 +Note. Error bars given here are the standard deviation of individual measurements and do not account for the 1.5 mas systematic +error on the absolute astrometric reference frame of NIRC2 due to the optical distortion correction for such a wide binary. +Relative photometry is given as the difference in magnitude ∆m ≡ mB − mA. +etry has a Gaussian-equivalent standard deviation of σ ≈ 0′′.5 +(Shimwell et al. 2022) yielding a 4σ discrepancy in position. +The Pan-STARRS1 z − y colour suggests that the source is a mid +M-dwarf whose zP1 = 21.06±0.06 mag places it at a photometric +distance of over 300 pc. This is well beyond the sensitivity hori- +zon of LOFAR for M-dwarfs’ cyclotron maser emission (Call- +ingham et al. 2021). Finally, the rotation rate implied by the radio +observations of 0.32 hr−1 is unusually large for a mid M-dwarf +(Popinchalk et al. 2021). For these reasons, we reject the associ- +ation between the radio source and PSO J154.7727+65.4978. +In order to compute CH4s−H colors for the two components +of WISEP J101905.63+652954.2 from Keck LGS AO imag- +ing, we used its IRTF/SpeX spectrum from 2010 May 27 UT +(Kirkpatrick et al. 2011) to measure integrated-light colors of +CH4s−H = −0.42 mag and J −H = −0.34 mag. Combined with +the 2MASS measurement of J = 16.589 ± 0.055 mag, these col- +ors give integrated-light photometry of H = 16.93±0.06 mag and +CH4s = 16.51±0.06 mag. Combined with our measured magni- +tude differences in CH4s and H, we find colors of CH4s − H = +−0.382 ± 0.013 mag and −0.481 ± 0.020 mag for the primary +and secondary. Using the spectral type-colour relation detailed +by Liu et al. (2008), we determine methane-photometry-based +spectral types of T5.5 ± 0.5 and T7.0 ± 0.5. +Article number, page 3 of 7 + +A&A proofs: manuscript no. main +0 +1 +2 +3 +4 +Frequency [1/hour] +−0.025 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +Lomb-Scargle power +FAR 0.1 +FAR 0.03 +FAR 0.01 +Sampling window +Data +Fig. 2. Lomb Scargle periodogram of the radio light curve from Fig. 1. +The dominant peak with a false alarm rate of under 3% is at 0.324 ± +0.033 hr−1. +3. Discussion +3.1. Mass and magnetic field estimates +We computed the combined-light bolometric luminosity of +WISEP J101905.63+652954.2 by direct integration of its unre- +solved optical to mid-infrared (MIR) spectral energy distribution +(SED). The assembled SED consists of available Pan-STARRS- +1 (PS1; Chambers et al. 2016) optical photometry (z, y), the +near-infrared (NIR) IRTF/SpeX prism spectrum, NIR photom- +etry from 2MASS (Cutri et al. 2003) and MKO (Best et al. +2021), and MIR photometry from the CatWISE catalog (W1 +and W2 bands; Eisenhardt et al. 2020; Marocco et al. 2021), +AllWISE catalog (W3 and W4 bands; Cutri et al. 2013), and +Spitzer/IRAC Channels 1 and 2 (Fazio et al. 2004). First, we +flux-calibrated the SpeX spectrum using the weighted average +of scale factors derived from PS1 y, 2MASS JHKs, and MKO +JHK photometry, assuming a systematic noise floor of 0.01 mag +for all the filters. We then integrated the flux-calibrated SpeX +spectrum to determine the NIR contribution to the bolometric +flux, accounting for the uncertainties in the spectral data points +and the flux calibration procedures. We determined the opti- +cal and MIR contributions to the bolometric flux by simultane- +ously fitting ATMO model atmospheres (Phillips et al. 2020) to +the PS1 and WISE photometry (computing synthetic photom- +etry from the models) and the SpeX spectrum (with the mod- +els degraded to the non-linear spectral resolution of the 0′′.5 +slit). We found the best-fitting ATMO model had Teff = 1000 +K and log g = 5.5 dex. Our final bolometric flux was found +by adding the NIR contribution to the integration of the model +outside the wavelength range of the SpeX spectrum. The uncer- +tainty in the optical+MIR contribution was obtained from the +standard deviation of the corresponding measurements derived +using the three model spectra adjacent in Teff and log g to the +best-fitting model. Using WISEPA J101905.63+652954.2’s par- +allax of 42.9 ± 1.8 mas, we calculated its bolometric luminosity +log(Lbol/L⊙) = −4.994 ± 0.063 dex. +To determine the mass of each component in the binary sys- +tem, we combined their luminosities and ages with the Saumon +& Marley (2008) (SM08) hybrid evolutionary model. Each com- +ponent’s luminosity is estimated using the bolometric luminosity +vs spectral type relations from Filippazzo et al. (2015). We find +that ≈70% of the luminosity is contributed by the T5.5 dwarf +and ≈30% is contributed by the T7 dwarf. For each object, we +adopt the field-age distribution from Dupuy & Liu (DL17 2017). +For our mass calculations, we use the Bayesian rejection sam- +pling technique described in Dupuy & Liu (2017). First, we draw +106 random (luminosity, age) samples from a uniform distribu- +tion spanning the bolometric luminosity range of the evolution- +ary model grid and the intersection of the DL17 age range and +the evolutionary model grid age range. Second, we compute the +probability of each sample based on the χ2 of the drawn lumi- +nosity with respect to the measured value and the likelihood of +drawing the sample’s age from the DL17 distribution. Third, we +randomly draw 106 uniform variates (u) distributed in the range +from 0 to 1 and reject any samples where p < u. The fourth and +final step is to linearly interpolate the evolutionary models (in +logarithmic space) at each accepted luminosity-age point to cal- +culate the corresponding mass. We find a mass of 41 ± 18 MJup +for the T5.5 component and 32 ± 16 MJup for the T7 component. +Armed with the mass and luminosity values, we can estimate +the magnetic fields of the two objects using so-called dynamo +scaling laws. We employed the ‘saturated dynamo’ scaling law +proposed by Christensen et al. (2009) that relates the magnetic +field to the heat flux and mean density of the brown dwarf. We +used the law in the form given by Reiners & Christensen (2010, +their equation 1). We also used their correction to estimate the +surface dipolar field from the field at the top of the dynamo as +predicted by the scaling law (Reiners & Christensen 2010, their +equation 2). Although the objects’ luminosities are have small +errors, the mass estimates and the normalising constant in the +scaling law have large fractional errors. To properly incorporate +these errors into the predicted magnetic field strength, we ran +a Monte-Carlo simulation where we drew the normalising con- +stant from a uniform distribution (Reiners et al. 2009, their equa- +tion 1), and the mass from a normal distribution. In each step +of the Monte Carlo run we interpolated the evolutionary mod- +els of Baraffe et al. (2003) to find the relationship between mass +and field strength for the measured luminosity (i.e. for different +ages). The resulting distribution of polar dipole field strengths +for the T5.5 and T7 objects had a mean and standard deviation +of 660 ± 300 G and 460 ± 210 G respectively. +The observed cyclotron maser emission itself places a lower +limit on the polar surface magnetic field strength of 51.4 G (cy- +clotron frequency at the mid-point of the LOFAR data’s radio +band). While this is consistent with the field estimates made +above, higher frequency observations are necessary to critically +test the dynamo scaling law. In what follows, we will leave the +polar surface field as a variable while normalising our equations +at B = 1 kG. +3.2. Energetics +WISEP J101905.63+652954.2 has not yet been detected at the +gigahertz-frequencies where quiescent incoherent synchrotron +emission is typically observed. A non-detection in the VLA Sky +Survey (Lacy et al. 2020) quick-look images yields a 3σ up- +per limit of 0.34 mJy in the 2–4 GHz band. Although incoher- +ent radio emission has widely been used as proxy for the en- +Article number, page 4 of 7 + +Vedantham et al.: Radio pulsation from new T-dwarf binary + + + + + + + + + + + + + + + + +CH4s + + + + + + + + + + + + + + + + 2015 Jan 15 +0.5" +J + + + + + + + + + + + + + + + + +H + + + + + + + + + + + + + + + + +CH4s + + + + + + + + + + + + + + + + 2022 Jan 24 +0.5" +J +Fig. 3. Contour plots of one typical individual exposure for each filter in which we obtained data. Contours are drawn in logarithmic intervals +from the peak flux down to 10% of the peak flux in each image. The images are all 1′′.5 across with North up. In 2015, despite the AO correction +deteriorating from 0′′.09 in the CH4s band to 0′′.13 in the J-band, the binary was still well resolved. We used the more precise differential magnitudes +from the higher-quality, and fully contemporaneous 2022 images in our analysis. +ergetics of magnetospheric and coronal emitters (Pineda et al. +2017; Leto et al. 2021; Benz & Guedel 1994), here we use the +pulsed radio emission to calculate the energetics of the auroral +electrons. We posit that the radio pulsations are due to beam- +ing combined with rotation and that the beam solid angle of the +radio emission is 1.6 sr — identical to that of Jupiter’s auroral +radio emission due to its magnetosphere–ionosphere coupling +(Zarka et al. 2004). The radio spectral luminosity for a pulse +flux density of 2 mJy (see Fig. 1) and a measured distance of +23.3 pc (Kirkpatrick et al. 2019) is then (2 mJy) × (23.3 pc)2 × +(1.6 sr) ≈ 1.7 × 1014 erg s−1 Hz−1. Let us further assume that the +total bandwidth of the radio emission is equal to the cyclotron +frequency at the surface of the object. Then the auroral radio +power is 4.6 × 1023 (B/kG) erg s−1. Assuming a 1% efficiency +in the conversion of the available auroral power to radio waves +(Zarka 2007; Lamy et al. 2011), we obtain an auroral power of +4.6×1025 (B/kG) erg s−1. For comparison, the auroral power out- +put of Jupiter is ∼ 1020 erg s−1 (Bhardwaj & Gladstone 2000), +and Turnpenney et al. (2017) predict auroral powers of up to +1026.5 erg s−1 (assuming the same 1% radio efficiency) for the Jo- +vian magnetosphere-ionosphere paradigm applied to ultra-cool +dwarfs. +3.3. Is binary interaction powering the radio emission? +Magnetic interaction between the two objects can accelerate +charges that eventually emit cyclotron maser radio emission, +as seen in the Jupiter-Io system (Goldreich & Lynden-Bell +1969; Neubauer 1998; Zarka 1998). The projected separation +of the two brown dwarfs in WISEP J101905.63+652954.2 is +9.9±0.4 au, given their parallactic distance of 23.3±1.0 pc (Kirk- +patrick et al. 2019). We explored the full range of orbital parame- +ters for the binary by fitting the two epochs of relative astrometry +from Section 2.2 with the orvara orbit analysis tool (Brandt et al. +2021). We used a prior on the total mass of 0.071 ± 0.033 M⊙ +based on our mass estimates from Section 3.1. As expected, the +orbital parameters are poorly constrained, but we can place 3σ +limits on the semimajor axis (> 5.2 au), period (> 30 yr), in- +clination (< 86◦), and eccentricity (< 0.96). The posterior dis- +tributions have medians and 1σ confidence intervals of 11+4 +−6 au, +160+120 +−130 yr, and 69+13 +−11 deg, but we caution that these are highly +influenced by the priors. (The eccentricity posterior is almost un- +changed from the uniform prior below the upper limit we quote.) +Based on the radio rotation rate of the emitter, its light cylin- +der is at a radial distance of about 3.4 au. Therefore, even if the +magnetospheres are not loaded with plasma (i.e. under force-free +electrodynamics), direct magnetic interaction between the two +dipolar magnetospheres is not possible and we must consider in- +terception by one brown dwarf of the Poynting flux radiated by +the other. The Poynting flux radiated by an oblique rotator (akin +to a Pulsar’s dipole emission) is of the order L ∼ B2 +0R6 +0Ω4/c3 +(Condon & Ransom 2016) where B0 is the surface magnetic +field, Ω is the angular rotation rate, and R0 is the object’s ra- +dius. For characteristic values of B0 = 103 G, R0 = 7 × 109 cm, +and Ω = 5.6 × 10−4 s−1, we get L ∼ 1020 erg s−1 which falls well +short of the value necessary to power the radio emission. +Next, consider a scenario where the magnetospheres are +loaded by plasma and drive a feeble wind. For simplicity, let +us assume that the two magnetospheres and their co-rotation +rates are similar. Due to the fast rotation, the balance between +the centrifugal force of the co-rotating plasma and magnetic +pressure must determine the structure of the magnetosphere in +this case (i.e. gravitational force can be safely neglected) and +the eventual Poynting flux. The centrifugal pressure felt by the +plasma is Fc = ρΩ2R2/2 where R is the radial distance, Ω is +the angular rotation rate and ρ = ρ(R) is the plasma density at +radius R. The magnetic pressure for a dipole at distance R is +FB = B2 +0R−6R6 +0/(8π) where R0 is the object’s radius and B0 is +the surface magnetic field strength. In our simple ‘toy model’, +at low radii, FB dominates enforcing co-rotation with a dipolar +field. This breaks at a critical radius when FB = FC. Beyond +this radius, we assume that the field lines open up into a Parker- +spiral type configuration. Note that FB = FC is equivalent to +saying that the co-rotation speed equals the local Alfvén speed. +The critical radius is therefore the so-called Alfvén point: +rA = +������ +B2 +0R6 +0 +4πΩ2ρ(rA) +������ +1/8 +, +(1) +In the open field zone, the azimuthal field dominates, falling +off with distance, R as R−1. We therefore assume B(R) += +B(rA)(R/rA)−1 where B(rA) = B0(rA/R0)−3. The brown dwarf +wind beyond rA is assumed to to have a radial flow speed, vr +equal to the co-rotation speed at rA as suggested for the Jovian +case by Hill et al. (1974). With these assumptions, the Poynting +luminosity can be readily computed as S = (B2/8π)×vr ×(4πR2) +at any closed surface of radius R > rA. The mass-loss rate is +given by ˙M = (4πr2 +A) × vr × ρ(rA). For parameters applicable to +WISEP J101905.63+652954.2 of R0 = 7 × 109 cm, B0 = 103 G, +Ω = 5.6 × 10−4 s−1, we find that the necessary Poynting lumi- +nosity of ≈ 1025.5 erg s−1 can be achieved with a mass-loss rate +of ≈ 25 tonnes per second. The corresponding Alfvén point is +at rA = 188R0. If instead we assume B0 = 100 G then we get +the necessary Poynting flux for +˙M ≈ 550 tonnes per second +and rA = 40R0. For comparison, Io’s volcanism is the princi- +pal source of Jovian magnetospheric plasma whose loss rate is +about 1 tonne per second. In any case, a significant fraction of +Article number, page 5 of 7 + +A&A proofs: manuscript no. main +the emitted Poynting flux must be intercepted by the magneto- +sphere of the companion for conversion of this Poynting flux +into radiation emission due to binary interaction. We therefore +conclude that while energetically feasible in principle, further +work on the precise details of the wind–wind interaction and the +source of mass-loss must be worked out to ascertain whether +this interaction could have powered the observed radio emission +from WISEP J101905.63+652954.2. +3.4. Auroral signatures +Regardless +of +the +veracity +of +the +interaction-powered +emission scenario, let us assume that at the emitter in +WISEP J101905.63+652954.2, an auroral mechanism similar +to that seen on Jupiter is at play. Such aurorae have also +been suggested as the radio emission mechanism in other +brown dwarfs and ultracool dwarfs (e.g. Hallinan et al. 2015; +Turnpenney et al. 2017). Jupiter’s aurorae emit compara- +ble amounts of power in the radio and Hα line (Bhardwaj +& Gladstone 2000; Zarka 1998). Assuming the same for +WISEP J101905.63+652954.2, we would anticipate an Hα lu- +minosity of 4.6 × 1023 (B/kG) erg s−1. Assuming a characteristic +line width of 6Å (Pineda et al. 2016), the expected Hα flux +density is ≈ 7 × 10−18 (B/kG) erg s−1 cm−2 Å−1. Based on the +optical spectrum or WISEP J101905.63+652954.2 presented +by Kirkpatrick et al. (2011), we derive a 2σ upper limit on the +Hα luminosity of 2.8 × 10−18 erg s−1 cm−2 Å−1. This suggests +that the surface magnetic field of WISEP J101905.63+652954.2 +is B ≲ 103 G which is broadly consistent with our magnetic +field estimate from §3.1. Nevertheless, we caution that it is +not possible to make definite statements on the magnetic field +strength because the radio and Hα efficiencies and the radio +beam solid angle can only be trusted to within an order of +magnitude. In conclusion, we find that the available data are +consistent with a Jupiter-like auroral process driving the radio +emission in a magnetosphere with a surface strength of order ap +kiloGauss. +4. Conclusions & Outlook +Magnetospheric emissions from the coldest brown dwarfs pro- +vide a rare glimpse into magnetism in the planetary mass +regime outside the solar system. Here we have presented our +second detection of a methane-bearing, T-type brown dwarf— +WISEP J101905.63+652954.2—with LOFAR at 144 MHz. The +radio emission is pulsed and periodic, from which we de- +rive a rotation rate of 0.32 ± 0.03 hr−1 (1σ bounds). We +have also presented infrared adaptive optics observations of +WISEP J101905.63+652954.2 that show it to be a T-dwarf bi- +nary with a separation of 9.9±0.4 au and spectral types T5.5±0.5 +and T7.0 ± 0.5, making it the first T-dwarf binary to be de- +tected in the radio band. We considered binarity as the cause +of the radio emission. We find that while energetically feasible +for mass-loss rates of ≳ 25 tonnes per second, precise details of +the interaction region must be studied before binary-interaction +can be posited as the probably cause of the emission. In this +regard, it is interesting to note that Kao & Sebastian Pineda +(2022) have suggested (based on detection rates and luminosi- +ties) that binary ultracool dwarfs may be more radio-loud than +their single counterparts. If this is true, then a radio-selection +as we have done here might reveal a population of close binary +brown dwarfs upon infrared follow-up observations, similar to +WISEP J101905.63+652954.2. +WISEP J101905.63+652954.2 is the first brown dwarf de- +tected at 144 MHz with the canonical periodic pulsed emission +profile similar to that seen in the cm-wave band and on Jupiter +at ν ≲ 40 MHz. Three previously detected T-dwarfs in the cm- +wave band have, unexpectedly, shown pulses up to 10 and/or 15 +GHz with no sign of a distinct high-frequency cut off (Kao et al. +2018). This suggests magnetic field strengths well in excess of +that anticipated by some dynamo scaling laws suggesting that the +laws need to be revised. However, it is also possible that by virtue +of a survey bias, the high frequency surveys have preferentially +detected a small population of T dwarfs that have anomalously +high field strengths possibly in smaller magnetic loops rather +than the large scale field predictions made from dynamo mod- +els. Because WISEP J101905.63+652954.2 was selected from a +144 MHz survey that does not have this selection bias, it will be +very interesting to see if its spectral cut-of continues to unex- +pected trend discovered by Kao et al. (2018). +We end by noting that WISEP J101905.63+652954.2 is the +second detected, and first pulsed, brown dwarf system found +in the ongoing LOFAR Two Metre Sky Survey. As demon- +strated by Vedantham et al. (2020), because the radio emission +is non-thermal in origin, radio surveys may be able to discover +a population of the coldest brown dwarfs that are too faint to +be found in canonical infrared surveys. 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HKV acknowledges funding from the Dutch Research Council (NWO) +for the project e-MAPS (project number Vi.Vidi.203.093) under the NWO tal- +ent scheme VIDI. JRC thanks NWO for support via the Talent Programme Veni +grant. LOFAR is the Low Frequency Array designed and constructed by AS- +TRON. It has observing, data processing, and data storage facilities in sev- +eral countries, that are owned by various parties (each with their own fund- +ing sources), and that are collectively operated by the ILT foundation under a +joint scientific policy. The ILT resources have benefitted from the following re- +cent major funding sources: CNRS-INSU, Observatoire de Paris and Université +d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation +Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ire- +land; NWO, The Netherlands; The Science and Technology Facilities Council, +UK. This research made use of the Dutch national e-infrastructure with the sup- +port of the SURF Cooperative (e-infra 180169) and the LOFAR e-infra group. +The Jülich LOFAR Long Term Archive and the German LOFAR network are +both coordinated and operated by the Jülich Supercomputing Centre (JSC), and +computing resources on the supercomputer JUWELS at JSC were provided by +the Gauss Centre for Supercomputing e.V. (grant CHTB00) through the John von +Neumann Institute for Computing (NIC). This research made use of the Uni- +versity of Hertfordshire high-performance computing facility and the LOFAR- +UK computing facility located at the University of Hertfordshire and supported +by STFC [ST/P000096/1], and of the Italian LOFAR IT computing infrastruc- +ture supported and operated by INAF, and by the Physics Department of Turin +University (under an agreement with Consorzio Interuniversitario per la Fisica +Spaziale) at the C3S Supercomputing Centre, Italy. Some of The data presented +herein were obtained at the W. M. Keck Observatory, which is operated as a sci- +entific partnership among the California Institute of Technology, the University +of California and the National Aeronautics and Space Administration. The Ob- +servatory was made possible by the generous financial support of the W. M. Keck +Foundation. The authors wish to recognise and acknowledge the very significant +cultural role and reverence that the summit of Maunakea has always had within +the indigenous Hawaiian community. We are most fortunate to have the opportu- +nity to conduct observations from this mountain. For the purpose of open access, +the author has applied a Creative Commons Attribution (CC BY) licence to any +Author Accepted Manuscript version arising from this submission. +Article number, page 7 of 7 + diff --git a/8NAzT4oBgHgl3EQfE_qO/content/tmp_files/load_file.txt b/8NAzT4oBgHgl3EQfE_qO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..67a1ba34f0a7808dabcd0283fc829a3f95839056 --- /dev/null +++ b/8NAzT4oBgHgl3EQfE_qO/content/tmp_files/load_file.txt @@ -0,0 +1,1014 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf,len=1013 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' main ©ESO 2023 January 4, 2023 Letter to the Editor Polarised radio pulsations from a new T dwarf binary H.' metadata={'source': 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France Received XXX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' accepted YYY ABSTRACT Brown dwarfs display Jupiter-like auroral phenomena such as magnetospheric Hα emission and coherent radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Coherent radio emission is a probe of magnetospheric acceleration mechanisms and provides a direct measurement of the magnetic field strength at the emitter’s location, both of which are difficult to access by other means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Observations of the coldest brown dwarfs (spectral types T and Y) are particularly interesting as their magnetospheric phenomena may be very similar to those in gas-giant exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Here we present 144 MHz radio and infrared adaptive optics observations of the brown dwarf WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 made using the LOFAR and Keck telescopes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The radio data shows pulsed highly circularly polarised emission which yields a rotation rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='03 hr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The infrared imaging reveals the source to be a binary with a projected separation of 423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 mas between components of spectral type T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 and T7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' With a simple “toy model” we show that the radio emission can in principle be powered by the interaction between the two dwarfs with a mass-loss rates of at least 25 times the Jovian value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 is interesting because it is the first pulsed methane dwarf detected in a low radio-frequency search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Unlike previous gigahertz-frequency searches that were only sensitive to objects with kiloGauss fields, our low-frequency search is sensitive to surface magnetic fields of ≈ 50 Gauss and above which might reveal the coldest radio-loud objects down to planetary mass-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Introduction High energy charges around brown dwarfs are expected to be created by auroral (or magnetospheric) processes akin to that seen on gas-giant planets, as opposed to coronal and chro- mospheric acceleration expected on stars (Nichols et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Williams 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Turnpenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This paradigm has been established based on highly circularly polarised and ro- tationally modulated radio pulses and Hα emission observed on brown dwarfs (Hallinan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2007, 2008, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Route & Wolszczan 2012, 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The radio emis- sion is of particular interest because it is expected to occur at the local cyclotron frequency, which in the non-relativistic limit, only depends on the ambient magnetic field strength (Melrose & Dulk 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Because Zeeman splitting observations become very challenging in such cold objects as brown dwarfs due to the lack of non-broadened spectral lines, radio observations may be the only viable technique to directly measure their magnetic field strengths and topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In addition, unlike rocky plan- ets, gas giants and brown dwarfs have predictable and relatively simple internal structures at depths where their magnetic field is expected to be generated (Chabrier & Baraffe 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This makes them ideal targets to test dynamo scaling laws (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=', Chris- tensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2009) that are likely applicable even in the exoplanet regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Despite concerted searches, radio detections of the cold- est brown dwarfs are rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The coldest, spectral type Y brown dwarfs have not been detected in the radio (Kao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' At the warmer spectral type T, four brown dwarfs have been de- tected in dedicated surveys at radio frequencies of 5 GHz and above (Route & Wolszczan 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Kao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Route & Wolszczan 2016b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Kao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2016, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Recently, we re- ported the first direct discovery of a brown dwarf made by virtue of its radio emission (Vedantham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2020) using the LO- FAR radio telescope (van Haarlem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2013) at 144 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Here we report our second discovery also made with LO- FAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 was originally discovered by Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2011) in Wide-field Infrared Survey Ex- plorer (WISE) data (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2010) and, using spectroscopic data, assigned optical and near-infrared spectral types of T7 and T6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Cold brown dwarfs share their radio phenomenology with Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The radio emission consists of two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' A quasi- quiescent component that is unpolarised or weakly polarised and a highly circularly polarised pulsed component that repeats at the rotation rate (Williams 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Antonova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Hallinan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Berger 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' However, the radio energetics of the detected brown dwarfs is orders of magnitude larger than that seen in Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This combined with a lack of detection of UV or H3+ from brown dwarfs (Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Gibbs & Fitzgerald 2022) suggest that the Jovian auroral energetics cannot be simply scaled to brown dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In any case, magnetic field lower lim- its derived from the pulsed component in three of the detected T dwarfs have been over a factor of three larger than the pre- dictions of leading dynamo scaling laws that can successfully predict the field strength of some solar system planets and low mass stars (Kao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This suggests that the model is inadequate, or it is also possible that by virtue of observing at Article number, page 1 of 7 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='01003v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='SR] 3 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' main high frequencies, the previous radio surveys were only sensitive to objects with anomalously large magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For instance, Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2009) predict a field strength of 103 Gauss for a 50 MJup brown dwarf with an age of 109 yr and a surface tem- perature of 1500 K (late-L dwarf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The corresponding peak cy- clotron frequency in its magnetosphere is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='8 GHz which will make such an object undetectable in a 5 GHz survey even if it were ‘typical’ of the predicted population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The 144 MHz LOFAR data can detect objects with surface field strengths as low as 50 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Therefore, the LOFAR-detected objects such as WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 are beginning to provide a more complete sample to critically test dynamo scaling laws over a larger range in magnetic field strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This paper is organised as follows: §2 presents details of the radio and infrared observations and the analysis of the radio light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In §3 we discuss the possible mechanisms driving the ra- dio emission, and present our conclusions and outlook in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Observations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' LOFAR 144 MHz observations WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 was discovered as part of our ongoing search (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Callingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2021) for stars, brown dwarfs, and exoplanets using data from the LOFAR Two Metre Sky Survey (LoTSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Shimwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Our methodology has typically involved searching for circularly polarised sources in deep 8 hr exposure LoTSS images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This is how we found Elegast, the first radio-selected brown dwarf (Vedantham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Because brown dwarf auroral emission is typically pulsed at the rotation period, we have since implemented a search al- gorithm to construct Stokes-V light curves on various time-bin widths and search for on-off and periodic pulsations from known brown dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Although we plan to conduct an untargeted search for such pulses over the Northern sky, we first validated our ap- proach by a targeted search of ten known T- and Y-type brown dwarfs which led to the discovery of Stokes-V radio pulsations from WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Our current pipeline takes in the standard calibrated visi- bilities from the LoTSS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We first subtract the LoTSS- detected sources from the visibilities using their direction depen- dent gains while retaining on-axis sources in the direction of the target for an additional round of self-calibration as described in van Weeren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We then modelled and subtracted these sources using their clean components from wsclean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We then imaged the target fields using wsclean (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2014) at a cadence of 4 minutes and extracted the light curves from these images after averaging over the available bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The light curve of WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 shows a statistically sig- nificant burst (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The on and flanking off-burst snapshot im- ages are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The figure also shows the light curve binned to a resolution of 16 min in red that reveals a hint of periodicity at around a 3 hr period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Polarised radio emission from planets and brown dwarfs is expected to have a periodic signature at the rotation period of the object due to beaming (akin to a light house).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' To ascertain the period signature in the light curve, we computed the Lomb- Scargle periodogram of the light curve using the astropy (As- tropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2013, 2018) implementation (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' To compute the significance of the periodogram peaks we computed the false alarm rate based on the bootstrap method de- scribed in VanderPlas (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We detect a dominant peak at a frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='324 hr−1 with a false alarm rate of under 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We compute an uncertainty in the peak’s location of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='033 hr−1 using the method prescribed in equation 52 of VanderPlas (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Keck/NIRC2 LGS AO We observed WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 on 2015 January 15 UT and 2022 January 24 UT with the facility imager NIRC2 at the Keck II telescope in concert with the laser guide star (LGS) adaptive optics (AO) system (van Dam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Wizinowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For tip-tilt correction, we used the star USNO- B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='0 1554-0140735, which is 23′′ away from the target and pro- vided flux to the tip-tilt sensor equivalent to R = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The wavefront sensor monitoring the LGS measured flux equivalent to a V = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 mag star in 2015 and V = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 mag in 2022, thanks to the intervening LGS upgrade (Chin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We obtained images with standard Maunakea Observatories filters in the J and H bands (Tokunaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2002) as well as CH4s, a medium- bandwidth filter centred on the H-band flux peak of T dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For each filter, we obtained between four and six dithered 180-s images in 2015 and 60-s images in 2022 while keeping the LGS fixed to the centre of NIRC2’s narrow camera (0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='01 pixel−1) field-of-view (10′′ × 10′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In 2015, the AO correction deterio- rated significantly as we collected data, and the quality of our H-band data set was too poor to be included in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We reduced our data using the same custom scripts as in our previous work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=', Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Dupuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2015), and examples of individual exposures are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We measured the separation, position angle (PA), and magnitude difference in individual exposures using three-component, two- dimensional Gaussians, and computed the means and standard deviations of measurements from individual exposures as the fi- nal measurements and their uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For our 2015 data, we used the astrometric calibration of Yelda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2010) to correct for nonlinear distortion, the orientation of NIRC2 (by subtracting 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='252), and the pixel scale (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='952±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='002 mas pixel−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Likewise, for our 2022 data we used the calibration of Service et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The resulting binary parameters are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Our relative astrometry is consistent within the errors at each epoch, and the repeated observations in J and CH4s filters show no significant change in flux ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' To compute the final relative astrometry at each epoch, we took the weighted average of our relative astrometry measure- ments and added a systematic error of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 mas to account for the uncertainty in the distortion corrections of NIRC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This gives separations of 423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 mas and 468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 mas and PAs of 161◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='71±0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='23 and 166◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='87±0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='20, in 2015 and 2022, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The observed motion of ≈ 7 mas yr−1 is much lower than the proper motion of the system (150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='1 mas yr−1) measured by Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2019), so we conclude the companion shares a common proper motion with the primary and is physically bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Our Keck images also showed a fainter point source ≈2′′ away from WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 at a position angle of ≈200◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We identified this source in the Pan-STARRS1 3π Survey catalog (Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2016) as PSO J154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='7727+65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='4978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' It is visible in stacked rizyP1 images and appears brightest in zP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Its (z − y)P1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='13 mag color (using stacked photometry) is far too blue to be a fainter, later-T or Y dwarf (Best et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2018), so we conclude this is a background star or galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Although this source is only about 2′′ from the nominal position of the radio detection, it is almost certainly not the source of the observed radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The high circular polarisation in the radio-band is inconsistent with an extragalactic origin, so we only need con- sider the Galactic stellar hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The absolute radio astrom- Article number, page 2 of 7 Vedantham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' : Radio pulsation from new T-dwarf binary 10h19m15s 10s 05s 00s 65±3003000 0000 2903000 0000 RA (J2000) Dec (J2000) Stokes V 10h19m15s 10s 05s 00s 65±3003000 0000 2903000 0000 RA (J2000) Dec (J2000) Stokes V 10h19m15s 10s 05s 00s 65±3003000 0000 2903000 0000 RA (J2000) Dec (J2000) Stokes V (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Panel (a) shows the Stokes-V radio lightcurve of WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 with a bin width of 4 minutes (black points with ±1σ errorbars) and 16 minutes (red curve with shaded ±1σ uncertainty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The point marked with the black square is a significant detection with a flux-density of −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='1(7) mJy whose Stokes-V image is shown in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Panels (b) and (d) show similar 4 min exposure images bracketing the integration show in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Keck LGS AO Relative Astrometry and Photometry of WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Epoch (MJD) Filter Separation (mas) PA (deg) ∆m (mag) 57037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5382 J 416 ± 7 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='06 57037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5246 CH4s 423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='489 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='019 59603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5270 J 467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='1 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='494 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='021 59603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5218 CH4s 467 ± 3 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='03 59603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5174 H 468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='7 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='579 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='013 Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Error bars given here are the standard deviation of individual measurements and do not account for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 mas systematic error on the absolute astrometric reference frame of NIRC2 due to the optical distortion correction for such a wide binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Relative photometry is given as the difference in magnitude ∆m ≡ mB − mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' etry has a Gaussian-equivalent standard deviation of σ ≈ 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 (Shimwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2022) yielding a 4σ discrepancy in position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The Pan-STARRS1 z − y colour suggests that the source is a mid M-dwarf whose zP1 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='06±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='06 mag places it at a photometric distance of over 300 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This is well beyond the sensitivity hori- zon of LOFAR for M-dwarfs’ cyclotron maser emission (Call- ingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Finally, the rotation rate implied by the radio observations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='32 hr−1 is unusually large for a mid M-dwarf (Popinchalk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For these reasons, we reject the associ- ation between the radio source and PSO J154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='7727+65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='4978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In order to compute CH4s−H colors for the two components of WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 from Keck LGS AO imag- ing, we used its IRTF/SpeX spectrum from 2010 May 27 UT (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2011) to measure integrated-light colors of CH4s−H = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='42 mag and J −H = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='34 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Combined with the 2MASS measurement of J = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='589 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='055 mag, these col- ors give integrated-light photometry of H = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='06 mag and CH4s = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='06 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Combined with our measured magni- tude differences in CH4s and H, we find colors of CH4s − H = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='382 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='013 mag and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='481 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='020 mag for the primary and secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Using the spectral type-colour relation detailed by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2008), we determine methane-photometry-based spectral types of T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 and T7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Article number, page 3 of 7 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' main 0 1 2 3 4 Frequency [1/hour] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='150 Lomb-Scargle power FAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='1 FAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='03 FAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='01 Sampling window Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Lomb Scargle periodogram of the radio light curve from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The dominant peak with a false alarm rate of under 3% is at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='324 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='033 hr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Mass and magnetic field estimates We computed the combined-light bolometric luminosity of WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 by direct integration of its unre- solved optical to mid-infrared (MIR) spectral energy distribution (SED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The assembled SED consists of available Pan-STARRS- 1 (PS1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2016) optical photometry (z, y), the near-infrared (NIR) IRTF/SpeX prism spectrum, NIR photom- etry from 2MASS (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2003) and MKO (Best et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2021), and MIR photometry from the CatWISE catalog (W1 and W2 bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Eisenhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Marocco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2021), AllWISE catalog (W3 and W4 bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2013), and Spitzer/IRAC Channels 1 and 2 (Fazio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' First, we flux-calibrated the SpeX spectrum using the weighted average of scale factors derived from PS1 y, 2MASS JHKs, and MKO JHK photometry, assuming a systematic noise floor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='01 mag for all the filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We then integrated the flux-calibrated SpeX spectrum to determine the NIR contribution to the bolometric flux, accounting for the uncertainties in the spectral data points and the flux calibration procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We determined the opti- cal and MIR contributions to the bolometric flux by simultane- ously fitting ATMO model atmospheres (Phillips et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2020) to the PS1 and WISE photometry (computing synthetic photom- etry from the models) and the SpeX spectrum (with the mod- els degraded to the non-linear spectral resolution of the 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 slit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We found the best-fitting ATMO model had Teff = 1000 K and log g = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Our final bolometric flux was found by adding the NIR contribution to the integration of the model outside the wavelength range of the SpeX spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The uncer- tainty in the optical+MIR contribution was obtained from the standard deviation of the corresponding measurements derived using the three model spectra adjacent in Teff and log g to the best-fitting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Using WISEPA J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2’s par- allax of 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='8 mas, we calculated its bolometric luminosity log(Lbol/L⊙) = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='994 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='063 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' To determine the mass of each component in the binary sys- tem, we combined their luminosities and ages with the Saumon & Marley (2008) (SM08) hybrid evolutionary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Each com- ponent’s luminosity is estimated using the bolometric luminosity vs spectral type relations from Filippazzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We find that ≈70% of the luminosity is contributed by the T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 dwarf and ≈30% is contributed by the T7 dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For each object, we adopt the field-age distribution from Dupuy & Liu (DL17 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For our mass calculations, we use the Bayesian rejection sam- pling technique described in Dupuy & Liu (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' First, we draw 106 random (luminosity, age) samples from a uniform distribu- tion spanning the bolometric luminosity range of the evolution- ary model grid and the intersection of the DL17 age range and the evolutionary model grid age range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Second, we compute the probability of each sample based on the χ2 of the drawn lumi- nosity with respect to the measured value and the likelihood of drawing the sample’s age from the DL17 distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Third, we randomly draw 106 uniform variates (u) distributed in the range from 0 to 1 and reject any samples where p < u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The fourth and final step is to linearly interpolate the evolutionary models (in logarithmic space) at each accepted luminosity-age point to cal- culate the corresponding mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We find a mass of 41 ± 18 MJup for the T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 component and 32 ± 16 MJup for the T7 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Armed with the mass and luminosity values, we can estimate the magnetic fields of the two objects using so-called dynamo scaling laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We employed the ‘saturated dynamo’ scaling law proposed by Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2009) that relates the magnetic field to the heat flux and mean density of the brown dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We used the law in the form given by Reiners & Christensen (2010, their equation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We also used their correction to estimate the surface dipolar field from the field at the top of the dynamo as predicted by the scaling law (Reiners & Christensen 2010, their equation 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Although the objects’ luminosities are have small errors, the mass estimates and the normalising constant in the scaling law have large fractional errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' To properly incorporate these errors into the predicted magnetic field strength, we ran a Monte-Carlo simulation where we drew the normalising con- stant from a uniform distribution (Reiners et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2009, their equa- tion 1), and the mass from a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In each step of the Monte Carlo run we interpolated the evolutionary mod- els of Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2003) to find the relationship between mass and field strength for the measured luminosity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' for different ages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The resulting distribution of polar dipole field strengths for the T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 and T7 objects had a mean and standard deviation of 660 ± 300 G and 460 ± 210 G respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The observed cyclotron maser emission itself places a lower limit on the polar surface magnetic field strength of 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='4 G (cy- clotron frequency at the mid-point of the LOFAR data’s radio band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' While this is consistent with the field estimates made above, higher frequency observations are necessary to critically test the dynamo scaling law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In what follows, we will leave the polar surface field as a variable while normalising our equations at B = 1 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Energetics WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 has not yet been detected at the gigahertz-frequencies where quiescent incoherent synchrotron emission is typically observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' A non-detection in the VLA Sky Survey (Lacy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2020) quick-look images yields a 3σ up- per limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='34 mJy in the 2–4 GHz band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Although incoher- ent radio emission has widely been used as proxy for the en- Article number, page 4 of 7 Vedantham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' : Radio pulsation from new T-dwarf binary CH4s 2015 Jan 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5" J H CH4s 2022 Jan 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5" J Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Contour plots of one typical individual exposure for each filter in which we obtained data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Contours are drawn in logarithmic intervals from the peak flux down to 10% of the peak flux in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The images are all 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 across with North up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In 2015, despite the AO correction deteriorating from 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='09 in the CH4s band to 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='13 in the J-band, the binary was still well resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We used the more precise differential magnitudes from the higher-quality, and fully contemporaneous 2022 images in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' ergetics of magnetospheric and coronal emitters (Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Leto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Benz & Guedel 1994), here we use the pulsed radio emission to calculate the energetics of the auroral electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We posit that the radio pulsations are due to beam- ing combined with rotation and that the beam solid angle of the radio emission is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 sr — identical to that of Jupiter’s auroral radio emission due to its magnetosphere–ionosphere coupling (Zarka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The radio spectral luminosity for a pulse flux density of 2 mJy (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 1) and a measured distance of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='3 pc (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2019) is then (2 mJy) × (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='3 pc)2 × (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 sr) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='7 × 1014 erg s−1 Hz−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Let us further assume that the total bandwidth of the radio emission is equal to the cyclotron frequency at the surface of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Then the auroral radio power is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 × 1023 (B/kG) erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Assuming a 1% efficiency in the conversion of the available auroral power to radio waves (Zarka 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Lamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2011), we obtain an auroral power of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6×1025 (B/kG) erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For comparison, the auroral power out- put of Jupiter is ∼ 1020 erg s−1 (Bhardwaj & Gladstone 2000), and Turnpenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2017) predict auroral powers of up to 1026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 erg s−1 (assuming the same 1% radio efficiency) for the Jo- vian magnetosphere-ionosphere paradigm applied to ultra-cool dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Is binary interaction powering the radio emission?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Magnetic interaction between the two objects can accelerate charges that eventually emit cyclotron maser radio emission, as seen in the Jupiter-Io system (Goldreich & Lynden-Bell 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Neubauer 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Zarka 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The projected separation of the two brown dwarfs in WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='4 au, given their parallactic distance of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='0 pc (Kirk- patrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We explored the full range of orbital parame- ters for the binary by fitting the two epochs of relative astrometry from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 with the orvara orbit analysis tool (Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We used a prior on the total mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='071 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='033 M⊙ based on our mass estimates from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' As expected, the orbital parameters are poorly constrained, but we can place 3σ limits on the semimajor axis (> 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 au), period (> 30 yr), in- clination (< 86◦), and eccentricity (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The posterior dis- tributions have medians and 1σ confidence intervals of 11+4 −6 au, 160+120 −130 yr, and 69+13 −11 deg, but we caution that these are highly influenced by the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (The eccentricity posterior is almost un- changed from the uniform prior below the upper limit we quote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=') Based on the radio rotation rate of the emitter, its light cylin- der is at a radial distance of about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='4 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Therefore, even if the magnetospheres are not loaded with plasma (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' under force-free electrodynamics), direct magnetic interaction between the two dipolar magnetospheres is not possible and we must consider in- terception by one brown dwarf of the Poynting flux radiated by the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The Poynting flux radiated by an oblique rotator (akin to a Pulsar’s dipole emission) is of the order L ∼ B2 0R6 0Ω4/c3 (Condon & Ransom 2016) where B0 is the surface magnetic field, Ω is the angular rotation rate, and R0 is the object’s ra- dius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For characteristic values of B0 = 103 G, R0 = 7 × 109 cm, and Ω = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 × 10−4 s−1, we get L ∼ 1020 erg s−1 which falls well short of the value necessary to power the radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Next, consider a scenario where the magnetospheres are loaded by plasma and drive a feeble wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For simplicity, let us assume that the two magnetospheres and their co-rotation rates are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Due to the fast rotation, the balance between the centrifugal force of the co-rotating plasma and magnetic pressure must determine the structure of the magnetosphere in this case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' gravitational force can be safely neglected) and the eventual Poynting flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The centrifugal pressure felt by the plasma is Fc = ρΩ2R2/2 where R is the radial distance, Ω is the angular rotation rate and ρ = ρ(R) is the plasma density at radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The magnetic pressure for a dipole at distance R is FB = B2 0R−6R6 0/(8π) where R0 is the object’s radius and B0 is the surface magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In our simple ‘toy model’, at low radii, FB dominates enforcing co-rotation with a dipolar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This breaks at a critical radius when FB = FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Beyond this radius, we assume that the field lines open up into a Parker- spiral type configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Note that FB = FC is equivalent to saying that the co-rotation speed equals the local Alfvén speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The critical radius is therefore the so-called Alfvén point: rA = ������ B2 0R6 0 4πΩ2ρ(rA) ������ 1/8 , (1) In the open field zone, the azimuthal field dominates, falling off with distance, R as R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We therefore assume B(R) = B(rA)(R/rA)−1 where B(rA) = B0(rA/R0)−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The brown dwarf wind beyond rA is assumed to to have a radial flow speed, vr equal to the co-rotation speed at rA as suggested for the Jovian case by Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' With these assumptions, the Poynting luminosity can be readily computed as S = (B2/8π)×vr ×(4πR2) at any closed surface of radius R > rA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The mass-loss rate is given by ˙M = (4πr2 A) × vr × ρ(rA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For parameters applicable to WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 of R0 = 7 × 109 cm, B0 = 103 G, Ω = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 × 10−4 s−1, we find that the necessary Poynting lumi- nosity of ≈ 1025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 erg s−1 can be achieved with a mass-loss rate of ≈ 25 tonnes per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The corresponding Alfvén point is at rA = 188R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' If instead we assume B0 = 100 G then we get the necessary Poynting flux for ˙M ≈ 550 tonnes per second and rA = 40R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For comparison, Io’s volcanism is the princi- pal source of Jovian magnetospheric plasma whose loss rate is about 1 tonne per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In any case, a significant fraction of Article number, page 5 of 7 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' main the emitted Poynting flux must be intercepted by the magneto- sphere of the companion for conversion of this Poynting flux into radiation emission due to binary interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We therefore conclude that while energetically feasible in principle, further work on the precise details of the wind–wind interaction and the source of mass-loss must be worked out to ascertain whether this interaction could have powered the observed radio emission from WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Auroral signatures Regardless of the veracity of the interaction-powered emission scenario, let us assume that at the emitter in WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2, an auroral mechanism similar to that seen on Jupiter is at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Such aurorae have also been suggested as the radio emission mechanism in other brown dwarfs and ultracool dwarfs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Hallinan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Turnpenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Jupiter’s aurorae emit compara- ble amounts of power in the radio and Hα line (Bhardwaj & Gladstone 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Zarka 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Assuming the same for WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2, we would anticipate an Hα lu- minosity of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='6 × 1023 (B/kG) erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Assuming a characteristic line width of 6Å (Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2016), the expected Hα flux density is ≈ 7 × 10−18 (B/kG) erg s−1 cm−2 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Based on the optical spectrum or WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 presented by Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2011), we derive a 2σ upper limit on the Hα luminosity of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='8 × 10−18 erg s−1 cm−2 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This suggests that the surface magnetic field of WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 is B ≲ 103 G which is broadly consistent with our magnetic field estimate from §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Nevertheless, we caution that it is not possible to make definite statements on the magnetic field strength because the radio and Hα efficiencies and the radio beam solid angle can only be trusted to within an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In conclusion, we find that the available data are consistent with a Jupiter-like auroral process driving the radio emission in a magnetosphere with a surface strength of order ap kiloGauss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Conclusions & Outlook Magnetospheric emissions from the coldest brown dwarfs pro- vide a rare glimpse into magnetism in the planetary mass regime outside the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Here we have presented our second detection of a methane-bearing, T-type brown dwarf— WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2—with LOFAR at 144 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The radio emission is pulsed and periodic, from which we de- rive a rotation rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='03 hr−1 (1σ bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We have also presented infrared adaptive optics observations of WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 that show it to be a T-dwarf bi- nary with a separation of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='4 au and spectral types T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5 and T7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='5, making it the first T-dwarf binary to be de- tected in the radio band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We considered binarity as the cause of the radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We find that while energetically feasible for mass-loss rates of ≳ 25 tonnes per second, precise details of the interaction region must be studied before binary-interaction can be posited as the probably cause of the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' In this regard, it is interesting to note that Kao & Sebastian Pineda (2022) have suggested (based on detection rates and luminosi- ties) that binary ultracool dwarfs may be more radio-loud than their single counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' If this is true, then a radio-selection as we have done here might reveal a population of close binary brown dwarfs upon infrared follow-up observations, similar to WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 is the first brown dwarf de- tected at 144 MHz with the canonical periodic pulsed emission profile similar to that seen in the cm-wave band and on Jupiter at ν ≲ 40 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Three previously detected T-dwarfs in the cm- wave band have, unexpectedly, shown pulses up to 10 and/or 15 GHz with no sign of a distinct high-frequency cut off (Kao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This suggests magnetic field strengths well in excess of that anticipated by some dynamo scaling laws suggesting that the laws need to be revised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' However, it is also possible that by virtue of a survey bias, the high frequency surveys have preferentially detected a small population of T dwarfs that have anomalously high field strengths possibly in smaller magnetic loops rather than the large scale field predictions made from dynamo mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Because WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 was selected from a 144 MHz survey that does not have this selection bias, it will be very interesting to see if its spectral cut-of continues to unex- pected trend discovered by Kao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We end by noting that WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 is the second detected, and first pulsed, brown dwarf system found in the ongoing LOFAR Two Metre Sky Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' As demon- strated by Vedantham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (2020), because the radio emission is non-thermal in origin, radio surveys may be able to discover a population of the coldest brown dwarfs that are too faint to be found in canonical infrared surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The pulsed emission from WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 therefore 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2021, ApJ, 916, 77 Reiners, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=', Basri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=', & Christensen, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2009, ApJ, 697, 373 Reiners, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' & Christensen, U.' metadata={'source': 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al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2016, PASP, 128, 095004 Shimwell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=', Hardcastle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=', Tasse, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=', et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2007, Planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=', 55, 598 Zarka, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=', Cecconi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=', & Kurth, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' 2004, Journal of Geophysical Research (Space Physics), 109, A09S15 Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Davy Kirkpatrick for making the Keck optical spectrum of WISEP J101905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='63+652954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='2 available to us in machine-readable format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' HKV acknowledges funding from the Dutch Research Council (NWO) for the project e-MAPS (project number Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='Vidi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='093) under the NWO tal- ent scheme VIDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' JRC thanks NWO for support via the Talent Programme Veni grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' LOFAR is the Low Frequency Array designed and constructed by AS- TRON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' It has observing, data processing, and data storage facilities in sev- eral countries, that are owned by various parties (each with their own fund- ing sources), and that are collectively operated by the ILT foundation under a joint scientific policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The ILT resources have benefitted from the following re- cent major funding sources: CNRS-INSU, Observatoire de Paris and Université d’Orléans, France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' BMBF, MIWF-NRW, MPG, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Science Foundation Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ire- land;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' NWO, The Netherlands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The Science and Technology Facilities Council, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This research made use of the Dutch national e-infrastructure with the sup- port of the SURF Cooperative (e-infra 180169) and the LOFAR e-infra group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The Jülich LOFAR Long Term Archive and the German LOFAR network are both coordinated and operated by the Jülich Supercomputing Centre (JSC), and computing resources on the supercomputer JUWELS at JSC were provided by the Gauss Centre for Supercomputing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' (grant CHTB00) through the John von Neumann Institute for Computing (NIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' This research made use of the Uni- versity of Hertfordshire high-performance computing facility and the LOFAR- UK computing facility located at the University of Hertfordshire and supported by STFC [ST/P000096/1], and of the Italian LOFAR IT computing infrastruc- ture supported and operated by INAF, and by the Physics Department of Turin University (under an agreement with Consorzio Interuniversitario per la Fisica Spaziale) at the C3S Supercomputing Centre, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Some of The data presented herein were obtained at the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Keck Observatory, which is operated as a sci- entific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The Ob- servatory was made possible by the generous financial support of the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Keck Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' The authors wish to recognise and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' We are most fortunate to have the opportu- nity to conduct observations from this mountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} +page_content=' Article number, page 7 of 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfE_qO/content/2301.01003v1.pdf'} diff --git a/8NAzT4oBgHgl3EQfSPtl/content/tmp_files/2301.01229v1.pdf.txt b/8NAzT4oBgHgl3EQfSPtl/content/tmp_files/2301.01229v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d74561574ed3ff198c8166ae4f36ee19e89edfe --- /dev/null +++ b/8NAzT4oBgHgl3EQfSPtl/content/tmp_files/2301.01229v1.pdf.txt @@ -0,0 +1,340 @@ +arXiv:2301.01229v1 [hep-lat] 3 Jan 2023 +Deconfinement in pure gauge SU(3) Yang-Mills theory: the +ghost propagator +Orlando Oliveira1,∗, Vítor Paiva1,∗∗, and Paulo Silva1,∗∗∗ +1CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal +Abstract. The ghost propagator in Landau gauge is studied at finite temperature +below and above Tc using lattice QCD simulations. For high temperatures, we +find that the ghost propagator is enhanced, compared to the confined phase. +The results suggest that the ghost propagator can be used to identify the phase +transition, similarly to the gluon propagator case. +1 Introduction +The QCD phase diagram has been the subject of several recent theoretical studies, motivated +by heavy ion experimental programs. At zero density, one expects a phase transition where +quarks and gluons become deconfined at high temperatures. The Polyakov loop L is the order +parameter for this transition: for temperatures below the critical temperature Tc, L = 0 and +quarks and gluons are confined inside hadrons. For pure gauge theories Tc = 270 MeV; the +inclusion of dynamical quarks lowers this value to Tc ∼ 170 MeV. +In QCD, propagators of fundamental fields encode information about non-perturbative +phenomena, such as confinement, deconfinement and chiral symmetry breaking. Following +our previous studies of the Landau gauge gluon [1, 2] and quark [3, 4] propagators at finite +temperature, here we study the behaviour of the ghost propagator in Landau gauge at finite +temperature. +2 Ghost Propagator +2.1 Setup +On the lattice, the computation of the ghost propagator relies on the inversion of a discretized +version of the Faddeev-Popov matrix. For details see, for example, [5]. +In order to evaluate the behaviour of the ghost propagator below and above the critical +temperature, a number of lattice ensembles were considered, covering a range of temperatures +from 121 MeV up to 486 MeV, as summarized in table 1, where Ls is the number of lattice +sites in any spatial direction, Lt is the number of lattice sites in the temporal direction and a +is the lattice spacing. The temperature is defined as T = 1/(aLt). Following previous works, +here we only consider the first Matsubara frequency. +∗e-mail: orlando@uc.pt +∗∗e-mail: vpaiva462@gmail.com +∗∗∗e-mail: psilva@uc.pt + +Table 1. Lattice setup. +Temp. (MeV) +β +Ls +Lt +a [fm] +1/a (GeV) +121 +6.0000 +64 +16 +0.1016 +1.943 +194 +6.0000 +64 +10 +0.1016 +1.943 +243 +6.0000 +64 +8 +0.1016 +1.943 +260 +6.0347 +68 +8 +0.09502 +2.0767 +265 +5.8876 +52 +6 +0.1243 +1.5881 +275 +6.0684 +72 +8 +0.08974 +2.1989 +324 +6.0000 +64 +6 +0.1016 +1.943 +366 +6.0684 +72 +6 +0.08974 +2.1989 +486 +6.0000 +64 +4 +0.1016 +1.943 +For each of the temperatures studied, we used a lattice ensemble of 100 configurations. +Since an “all-to-all” propagatorwould be computationally extremely costly, two point sources +are considered for each configuration, one at the origin of the lattice, (0, 0, 0, 0), and one at +the lattice’s spatial midpoint, (Ls/2, Ls/2, Ls/2, 0). A simple average over the two is taken in +order to mimic an “all-to-all” propagator with “point-to-all” propagators. +In order to account for lattice artefacts for large momenta, the (physical) momenta above +1 GeV were subject to a cylindrical cut [6] where only momenta whose distance, d, from the +lattice’s diagonal was such that d a < 4 (2π/Ls) were considered in the final data – that is, +momenta less than four spatial units away from the lattice’s diagonal, (p, p, p, 0). +The propagators pertaining to different temperatures were renormalized at µ = 4 GeV, by +imposing G(µ2) = 1/µ2. In order to do so, a fit was performed to the propagators, with the +functional form +G(p2) = +b + cp2 +p4 + dp2 + e +, +(1) +where b, c, d and e are adjustable parameters. +2.2 Temperature Dependence +The effect of temperature in the ghost propagator for all momentum range is exhibited in +figures 1 and 2. Note that our results are similar to previous results using quenched ensembles +with smaller lattice volumes [7]. +The distinction between the behaviour below and above the critical temperature is only +made clear at lower values of the momenta, as was also observed for the gluon propagator. +Figure 2 zooms in on the infrared (IR) region of the ghost propagator, where the enhancement +of the propagator above Tc, relative to the confined case, is visible. Below the critical temper- +ature, the propagators for the different temperatures are compatible within statistical errors. +As Figure 3 further illustrates for the four lowest accessible momenta, the enhancement effect +rapidly decreases as the momentum increases and the two regimes become indistinguishable +for high momenta. +2.3 Z3 Dependence +On the lattice, gauge configurations related to each other through a center (or Z3) transforma- +tion are equivalent. The Wilson gauge action is invariant under a center transformation, which +consists in the multiplication of all time links in a constant temporal hyperplane, x4 = const, + +0 +1 +2 +3 +4 +5 +6 +7 +8 +p(GeV) +0,01 +0,1 +1 +10 +100 +G(p +2) +T = 121 MeV +T = 194 MeV +T = 243 MeV +T = 260 MeV +T = 265 MeV +T = 275 MeV +T = 324 MeV +T = 366 MeV +T = 486 MeV +Ghost Propagator at finite temperature +Renormalized at 4 GeV +Figure 1. Renormalized ghost propagator at finite temperature. +0 +0,2 +0,4 +0,6 +0,8 +1 +p(GeV) +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +G(p +2) +T = 121 MeV +T = 194 MeV +T = 243 MeV +T = 260 MeV +T = 265 MeV +T = 275 MeV +T = 324 MeV +T = 366 MeV +T = 486 MeV +Ghost Propagator at finite temperature +Renormalized at 4 GeV +Figure 2. Renormalized ghost propagator at finite temperature in the IR region. +by an element z of the center (or Z3) group, +Z3 = {e−i 2π +3 , 1, ei 2π +3 } . +(2) +The symmetry holds for closed loops like the Wilson loop. The Polyakov loop, L(⃗x), however, +is not invariant under such a transformation, L(⃗x) → zL(⃗x). It thus constitutes an order +parameter for the deconfinement phase transition. Below Tc, center symmetry holds and +⟨L⟩ = 0; above Tc, center symmetry is spontaneously broken, the Z3 sectors are not equally +populated and ⟨L⟩ � 0. +Previous works have shown that the gluon [2] and quark propagators [4] are sensitive +to the Z3 sector of the gauge configurations. Our preliminary results suggest that the ghost +propagator is also sensitive to the Z3 sector above Tc. Figure 4 shows the IR region of two +lattice simulations with Ls = 72 and Lt = 8 with β = 6.058 (left-hand panel) and β = 6.066 +(right-hand panel). The results show that the ghost propagator behaves differently below and + +100 +200 +300 +400 +500 +T (MeV) +75 +80 +85 +90 +G(p +2) +Ghost Propagator as a function of temperature +p = 191 MeV +100 +200 +300 +400 +500 +T (MeV) +33 +36 +39 +42 +G(p +2) +Ghost Propagator as a function of temperature +p = 270 MeV +100 +200 +300 +400 +500 +T (MeV) +20 +22 +24 +G(p +2) +Ghost Propagator as a function of temperature +p = 330 MeV +100 +200 +300 +400 +500 +T (MeV) +14 +15 +16 +17 +G(p +2) +Ghost Propagator as a function of temperature +p = 381 MeV +Figure 3. Ghost propagator as a function of temperature for p = 191 MeV (top left panel), p = 270 +MeV (top right panel), p = 330 MeV (left bottom panel) and p = 381 MeV (right bottom panel). The +red vertical line indicates the critical temperature Tc. +0 +0,2 +0,4 +0,6 +0,8 +1 +p (GeV) +0 +20 +40 +60 +80 +G(p +2) +sector -1 +sector 0 +sector 1 +0 +0,2 +0,4 +0,6 +0,8 +1 +p (GeV) +0 +20 +40 +60 +80 +G(p +2) +sector -1 +sector 0 +sector 1 +Figure 4. Ghost propagator’s sector dependence below Tc (left-hand panel at T = 270 MeV) and above +Tc (right-hand panel at T = 274 MeV). +above Tc, with a suppression of the ±1 sectors relative to the 0 sector for the deconfined phase. +As we found previously for the gluon propagator [2], the ±1 sectors are indistinguishable +above Tc. + +3 Conclusions and outlook +In this paper we study the Landau gauge ghost propagator at finite temperature using lattice +simulations. We found an enhancement of the ghost form factor above the critical tempera- +ture Tc, already found in previous SU(3) studies on smaller volumes [7]. Note that early +SU(2) studies concluded in favour of a nearly independent ghost propagator with the temper- +ature [8]. We also show preliminary results for the Z3 dependence of the ghost propagator. +Although the propagators in the various sectors are indistinguishable below Tc, we found a +suppression, above Tc, of the ±1 sectors in comparison with the 0 sector. However, in the +deconfined phase the ±1 sectors are still compatible within errors. +We are currently extending the study of the Z3 dependence for other temperatures. In the +near future we also plan to study the QCD propagators at finite temperature using dynamical +configurations. +Acknowledgements +This work was partly supported by the FCT – Fundação para a Ciência e a Tecnolo- +gia, I.P., under Projects Nos. +UIDB/04564/2020, UIDP/04564/2020 and CERN/FIS- +COM/0029/2017. P. J. S. acknowledges financial support from FCT (Portugal) under Con- +tract No. CEECIND/00488/2017. The authors acknowledge the Laboratory for Advanced +Computing at the University of Coimbra (http://www.uc.pt/lca) for providing access to the +HPC resource Navigator. +References +[1] P.J. Silva, O. Oliveira, P. Bicudo, N. Cardoso, Phys. Rev. D 89, 074503 (2014), +1310.5629 +[2] P.J. Silva, O. Oliveira, Phys. Rev. D 93, 114509 (2016), 1601.01594 +[3] O. Oliveira, P.J. Silva, Eur. Phys. J. C 79, 793 (2019), 1903.00263 +[4] P.J. Silva, O. Oliveira, PoS LATTICE2019, 047 (2020), 1912.13061 +[5] A. Cucchieri, D. Dudal, T. Mendes, O. Oliveira, M. Roelfs, P.J. Silva, PoS LAT- +TICE2018, 252 (2018), 1811.11521 +[6] D.B. Leinweber, J.I. Skullerud, A.G. Williams, C. Parrinello (UKQCD), Phys. Rev. D +60, 094507 (1999), [Erratum: Phys.Rev.D 61, 079901 (2000)], hep-lat/9811027 +[7] R. Aouane, V.G. Bornyakov, E.M. Ilgenfritz, V.K. Mitrjushkin, M. Müller-Preussker, +A. Sternbeck, Phys. Rev. D 85, 034501 (2012) +[8] A. Cucchieri, A. Maas, T. Mendes, Phys. Rev. D 75, 076003 (2007), hep-lat/0702022 + diff --git a/8NAzT4oBgHgl3EQfSPtl/content/tmp_files/load_file.txt b/8NAzT4oBgHgl3EQfSPtl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ac7539a5ffae480d67a52c22939c4b7b09d6064 --- /dev/null +++ b/8NAzT4oBgHgl3EQfSPtl/content/tmp_files/load_file.txt @@ -0,0 +1,226 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf,len=225 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='01229v1 [hep-lat] 3 Jan 2023 Deconfinement in pure gauge SU(3) Yang-Mills theory: the ghost propagator Orlando Oliveira1,∗, Vítor Paiva1,∗∗, and Paulo Silva1,∗∗∗ 1CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' The ghost propagator in Landau gauge is studied at finite temperature below and above Tc using lattice QCD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' For high temperatures, we find that the ghost propagator is enhanced, compared to the confined phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' The results suggest that the ghost propagator can be used to identify the phase transition, similarly to the gluon propagator case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' 1 Introduction The QCD phase diagram has been the subject of several recent theoretical studies, motivated by heavy ion experimental programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' At zero density, one expects a phase transition where quarks and gluons become deconfined at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' The Polyakov loop L is the order parameter for this transition: for temperatures below the critical temperature Tc, L = 0 and quarks and gluons are confined inside hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' For pure gauge theories Tc = 270 MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' the inclusion of dynamical quarks lowers this value to Tc ∼ 170 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' In QCD, propagators of fundamental fields encode information about non-perturbative phenomena, such as confinement, deconfinement and chiral symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Following our previous studies of the Landau gauge gluon [1, 2] and quark [3, 4] propagators at finite temperature, here we study the behaviour of the ghost propagator in Landau gauge at finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' 2 Ghost Propagator 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='1 Setup On the lattice, the computation of the ghost propagator relies on the inversion of a discretized version of the Faddeev-Popov matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' For details see, for example, [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' In order to evaluate the behaviour of the ghost propagator below and above the critical temperature, a number of lattice ensembles were considered, covering a range of temperatures from 121 MeV up to 486 MeV, as summarized in table 1, where Ls is the number of lattice sites in any spatial direction, Lt is the number of lattice sites in the temporal direction and a is the lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' The temperature is defined as T = 1/(aLt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Following previous works, here we only consider the first Matsubara frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' ∗e-mail: orlando@uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='pt ∗∗e-mail: vpaiva462@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='com ∗∗∗e-mail: psilva@uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='pt Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Lattice setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' (MeV) β Ls Lt a [fm] 1/a (GeV) 121 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='0000 64 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='1016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='943 194 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='0000 64 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='1016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='943 243 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='0000 64 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='1016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='943 260 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='0347 68 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='09502 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='0767 265 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='8876 52 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='1243 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='5881 275 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='0684 72 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='08974 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='1989 324 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='0000 64 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='1016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='943 366 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='0684 72 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='08974 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='1989 486 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='0000 64 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='1016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='943 For each of the temperatures studied, we used a lattice ensemble of 100 configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Since an “all-to-all” propagatorwould be computationally extremely costly, two point sources are considered for each configuration, one at the origin of the lattice, (0, 0, 0, 0), and one at the lattice’s spatial midpoint, (Ls/2, Ls/2, Ls/2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' A simple average over the two is taken in order to mimic an “all-to-all” propagator with “point-to-all” propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' In order to account for lattice artefacts for large momenta, the (physical) momenta above 1 GeV were subject to a cylindrical cut [6] where only momenta whose distance, d, from the lattice’s diagonal was such that d a < 4 (2π/Ls) were considered in the final data – that is, momenta less than four spatial units away from the lattice’s diagonal, (p, p, p, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' The propagators pertaining to different temperatures were renormalized at µ = 4 GeV, by imposing G(µ2) = 1/µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' In order to do so, a fit was performed to the propagators, with the functional form G(p2) = b + cp2 p4 + dp2 + e , (1) where b, c, d and e are adjustable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='2 Temperature Dependence The effect of temperature in the ghost propagator for all momentum range is exhibited in figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Note that our results are similar to previous results using quenched ensembles with smaller lattice volumes [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' The distinction between the behaviour below and above the critical temperature is only made clear at lower values of the momenta, as was also observed for the gluon propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Figure 2 zooms in on the infrared (IR) region of the ghost propagator, where the enhancement of the propagator above Tc, relative to the confined case, is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Below the critical temper- ature, the propagators for the different temperatures are compatible within statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' As Figure 3 further illustrates for the four lowest accessible momenta, the enhancement effect rapidly decreases as the momentum increases and the two regimes become indistinguishable for high momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='3 Z3 Dependence On the lattice, gauge configurations related to each other through a center (or Z3) transforma- tion are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' The Wilson gauge action is invariant under a center transformation, which consists in the multiplication of all time links in a constant temporal hyperplane, x4 = const, 0 1 2 3 4 5 6 7 8 p(GeV) 0,01 0,1 1 10 100 G(p 2) T = 121 MeV T = 194 MeV T = 243 MeV T = 260 MeV T = 265 MeV T = 275 MeV T = 324 MeV T = 366 MeV T = 486 MeV Ghost Propagator at finite temperature Renormalized at 4 GeV Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Renormalized ghost propagator at finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' 0 0,2 0,4 0,6 0,8 1 p(GeV) 0 10 20 30 40 50 60 70 80 90 G(p 2) T = 121 MeV T = 194 MeV T = 243 MeV T = 260 MeV T = 265 MeV T = 275 MeV T = 324 MeV T = 366 MeV T = 486 MeV Ghost Propagator at finite temperature Renormalized at 4 GeV Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Renormalized ghost propagator at finite temperature in the IR region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' by an element z of the center (or Z3) group, Z3 = {e−i 2π 3 , 1, ei 2π 3 } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' (2) The symmetry holds for closed loops like the Wilson loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' The Polyakov loop, L(⃗x), however, is not invariant under such a transformation, L(⃗x) → zL(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' It thus constitutes an order parameter for the deconfinement phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Below Tc, center symmetry holds and ⟨L⟩ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' above Tc, center symmetry is spontaneously broken, the Z3 sectors are not equally populated and ⟨L⟩ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Previous works have shown that the gluon [2] and quark propagators [4] are sensitive to the Z3 sector of the gauge configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Our preliminary results suggest that the ghost propagator is also sensitive to the Z3 sector above Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Figure 4 shows the IR region of two lattice simulations with Ls = 72 and Lt = 8 with β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='058 (left-hand panel) and β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='066 (right-hand panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' The results show that the ghost propagator behaves differently below and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='T (MeV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='G(p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='Ghost Propagator as a function of temperature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='p = 191 MeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='T (MeV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='G(p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='Ghost Propagator as a function of temperature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='p = 270 MeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='T (MeV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='G(p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='Ghost Propagator as a function of temperature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='p = 330 MeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='T (MeV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='G(p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='Ghost Propagator as a function of temperature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='p = 381 MeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Ghost propagator as a function of temperature for p = 191 MeV (top left panel), p = 270 MeV (top right panel), p = 330 MeV (left bottom panel) and p = 381 MeV (right bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' The red vertical line indicates the critical temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' 0 0,2 0,4 0,6 0,8 1 p (GeV) 0 20 40 60 80 G(p 2) sector -1 sector 0 sector 1 0 0,2 0,4 0,6 0,8 1 p (GeV) 0 20 40 60 80 G(p 2) sector -1 sector 0 sector 1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Ghost propagator’s sector dependence below Tc (left-hand panel at T = 270 MeV) and above Tc (right-hand panel at T = 274 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' above Tc, with a suppression of the ±1 sectors relative to the 0 sector for the deconfined phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' As we found previously for the gluon propagator [2], the ±1 sectors are indistinguishable above Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' 3 Conclusions and outlook In this paper we study the Landau gauge ghost propagator at finite temperature using lattice simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' We found an enhancement of the ghost form factor above the critical tempera- ture Tc, already found in previous SU(3) studies on smaller volumes [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Note that early SU(2) studies concluded in favour of a nearly independent ghost propagator with the temper- ature [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' We also show preliminary results for the Z3 dependence of the ghost propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Although the propagators in the various sectors are indistinguishable below Tc, we found a suppression, above Tc, of the ±1 sectors in comparison with the 0 sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' However, in the deconfined phase the ±1 sectors are still compatible within errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' We are currently extending the study of the Z3 dependence for other temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' In the near future we also plan to study the QCD propagators at finite temperature using dynamical configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Acknowledgements This work was partly supported by the FCT – Fundação para a Ciência e a Tecnolo- gia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=', under Projects Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' UIDB/04564/2020, UIDP/04564/2020 and CERN/FIS- COM/0029/2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} 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+page_content=' Maas, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Mendes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} +page_content=' D 75, 076003 (2007), hep-lat/0702022' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'} diff --git a/8dE5T4oBgHgl3EQfQg7H/content/tmp_files/2301.05514v1.pdf.txt b/8dE5T4oBgHgl3EQfQg7H/content/tmp_files/2301.05514v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ba8aba5fbfb5086bb5eec7792c12e26a0baca378 --- /dev/null +++ b/8dE5T4oBgHgl3EQfQg7H/content/tmp_files/2301.05514v1.pdf.txt @@ -0,0 +1,383 @@ +Primal-Dual Cops and Robber +Minh Tuan Ha � +Karlsruhe Institute of Technology, Germany +Paul Jungeblut � +Karlsruhe Institute of Technology, Germany +Torsten Ueckerdt � +Karlsruhe Institute of Technology, Germany +Abstract +Cops and Robber is a family of two-player games played on graphs in which one player controls a +number of cops and the other player controls a robber. In alternating turns, each player moves (all) +his/her figures. The cops try to capture the robber while the latter tries to flee indefinitely. In this +paper we consider a variant of the game played on a planar graph where the robber moves between +adjacent vertices while the cops move between adjacent faces. The cops capture the robber if they +occupy all incident faces. We prove that a constant number of cops suffices to capture the robber on +any planar graph of maximum degree ∆ if and only if ∆ ≤ 4. +2012 ACM Subject Classification Mathematics of computing → Discrete mathematics → Combi- +natorics → Combinatoric problems +Keywords and phrases Cops and robber, planar graph, dual graph +1 +Introduction +Cops and Robber is probably the most classical combinatorial pursuit-evasion game on graphs. +The robber models an intruder in a network that the cops try to capture. Two players play +with complete information on a fixed finite graph G = (V, E). The cop player controls a set +of k cops, each occupying a vertex of G (possibly several cops on the same vertex), while +the robber player controls a single robber that also occupies a vertex of G. The players +take alternating turns, where the cop player in his turn can decide for each cop individually +whether to stay at its position or move the cop along an edge of G onto an adjacent vertex. +Similarly, the robber player on her turn can leave the robber at its position or move it along +an edge of G. The cop player starts by choosing starting positions for his k cops and wins +the game as soon as at least one cop occupies the same vertex as the robber, i.e., when +the robber is captured. The robber player, seeing the cops positions, chooses the starting +position for her robber and wins if she can avoid capture indefinitely. The least integer k for +which, assuming perfect play on either side, k cops can always capture the robber, is called +the cop number of G, usually denoted by c(G). +In this paper, we introduce Primal-Dual Cops and Robber which is played on a plane +graph G, i.e., with a fixed plane embedding. Here, the cops occupy the faces of G and can +move between adjacent faces (i.e., faces that share an edge), while the robber still moves +along edges between adjacent vertices of G. In this game, the robber is captured if every +face incident to the robber’s vertex is occupied by at least one cop. Analogously, we call the +least integer k for which k cops can always capture the robber in the Primal-Dual Cops and +Robber game the primal-dual cop number of G and denote it by c∗(G). +An obvious lower bound for c∗(G) is the maximum number of faces incident to any vertex +in G: The robber can choose such a vertex as its start position and just stay there indefinitely +(note that there is no zugzwang, i.e., no obligation to move during ones turn). In particular, +if G has maximum degree ∆(G) and there exists a vertex v with deg(v) = ∆(G), which is +not a cut-vertex, then c∗(G) ≥ ∆(G). E.g., c∗(K2,n) = ∆(K2,n) = n for any n ≥ 2. +arXiv:2301.05514v1 [math.CO] 13 Jan 2023 + +2 +Primal-Dual Cops and Robber +Our contribution. +We investigate, whether the primal-dual cop number c∗(G) is bounded +in terms of ∆(G) for all plane graphs G. The answer is ‘Yes’ if ∆(G) ≤ 4 and ‘No’ otherwise. +▶ Theorem 1. Each of the following holds. +1. For every plane graph G with ∆(G) ≤ 3 we have c∗(G) ≤ 3. +2. For every plane graph G with ∆(G) ≤ 4 we have c∗(G) ≤ 12. +3. For some n-vertex plane graphs G with ∆(G) = 5 we have c∗(G) = Ω +�� +log(n) +� +. +Related work. +Let us just briefly mention that Cops and Robber was introduced by +Nowakowski and Winkler [10] and Quillot [12] for one cop and Aigner and Fromme [1] for k +cops 40 years ago. Since then numerous results and variants were presented, see e.g., [2, 3]. +Perhaps most similar to our new variant are the recent surrounding variant of Burgess et +al. [5] with vertex-cops and the containment variant of Cryster et al. [6, 11] with edge-cops. +In these variants the robber is captured if every adjacent vertex, respectively every incident +edge, is occupied by a cop. The smallest number of cops that always suffices for any planar +graph G is 3 in the classical variant [1], 7 in the surrounding variant [4], 7∆(G) in the +containment variant [6] and 3 when both, cops and robber, move on edges [7]. +2 +Cops win always if the maximum degree is at most four +We start with an observation that simplifies the proofs of items 1 and 2 in Theorem 1. +▶ Observation 2. Let the robber be on a vertex u with a neighbor v of degree 1. Then the +robber is never required to move to v to evade the cops. +This is true because the set of faces required to capture the robber at v is a subset of the +faces required to capture him at u. Further, his only possible moves at v are either staying +there or moving back to u. As there is no zugzwang, he could just stay at u all along. +In both of the following proofs we assume that the graph contains only degree-3-vertices +(respectively degree-4-vertices) and degree-1-vertices. This can always be achieved by adding +leaves to vertices not yet having the correct degree. +Proof of item 1 in Theorem 1. We give a winning strategy for three cops c1, c2, c3 in a +planar graph G with ∆(G) ≤ 3. First the cops choose arbitrary faces to start on. Then the +robber chooses its start vertex u, which we assume to be of degree 3 by Observation 2 (it +is trivial to capture him if all vertices have degree 1). Let ∠u +1, ∠u +2, ∠u +3 be the three angles +incident to u. We denote the face containing an angle ∠ by f(∠) and define for each cop ci a +target face fi, i = 1, 2, 3. Initially we set fi = f(∠u +i ). The goal of each cop is to reach his +target face, thereby capturing the robber when all three cops arrive. If the robber moves, +each cop updates his target face. Our strategy guarantees that the total distance of all three +cops to their targets faces decreases over time, so it reaches zero after finitely many turns. +Clearly, in every game the robber has to move at some point to avoid being captured. +Assume that the robber moves from vertex u to vertex v (both of degree 3 by Observation 2). +Without loss of generality the angles around u and v are labeled as in Figure 1 with fi = f(∠u +i ) +being the current target face of cop ci, i = 1, 2, 3. +First assume that c3 (or symmetrically c2) has not reached his target face yet. In this +case we assign the new target faces f1 = f(∠v +1), f2 = f(∠v +2) and f3 = f(∠v +3). Note that +for i = 1, 2 faces f(∠u +i ) and f(∠v +i ) are adjacent, so cop ci can keep his distance to his target +face unchanged (or even decrease it) during his next turn. Further note that f(∠u +3) = f(∠v +3), + +M. T. Ha, P. Jungeblut and T. Ueckerdt +3 +̸ +u +1 +̸ +u +2 +̸ +u +3 +̸ +v +1 +̸ +v +2 +̸ +v +3 +u +v +w +Figure 1 Labeling of the angles for a robber move from u to v (and possibly further to w). +v +u +̸ +u +1 +̸ +u +2 +̸ +u +3 +̸ +v +1 +̸ +v +2 +̸ +v +3 +̸ +u +4 +̸ +v +4 +Figure 2 A vertex cop and its four accompanying face-cops moving from u to v. +so cop c3 can even decrease his distance by one during the next turn. Thus the total distance +of the three cops to their target faces decreased by at least one. +It remains the case that c2 and c3 have already reached their target faces (but c1 did not, +as the game would be over otherwise). In this case we move c1 one step towards his target +face f1 = f(∠u +1) and c2, c3 both to f(∠v +2). Now its the robber’s turn again. If she does not +move, we assign target faces fi = f(∠v +i ), i = 1, 2, 3, and the total distance decreases after the +cops’ next turn. If she moves back to u, we assign target faces fi = f(∠u +i ), i = 1, 2, 3, and +the total distance decreases after the cops’ next turn. The last possibility for the robber is to +move towards another neighbor w of v, see Figure 1. Then we assign f1 = f(∠v +1) and f2, f3 +to be the faces containing the other two angles at w. In their next turn, c2 and c3 can again +reach their target faces, while c1 can decrease his distance to his target face f(∠v +1) by one +compared to the initial situation with the robber at vertex u. Again, the total distance is +decreased, which concludes the proof. +◀ +To prove item 2 in Theorem 1, we reduce our Primal-Dual Cops and Robber to the +classical Cops and Robber with cops on vertices of G and then use a result from the literature. +▶ Lemma 3. In a plane graph G with ∆(G) ≤ 4, four face-cops can simulate a vertex-cop. +Proof. Let c be a vertex-cop starting at a vertex u ∈ V (G) with up to four incident angles ∠u +i +(for i ∈ {1, 2, 3, 4}). We place four face-cops on the (up to) four faces f(∠u +i ). If the vertex-cop +moves to an adjacent vertex v, the four face cops around it can in one step also move to +faces containing the angles incident to v, see Figure 2 for the case that u and v both have +degree 4. For vertices of degree less then 4 it only gets easier for the face-cops. +◀ +An immediate corollary of Lemma 3 is that c∗(G) ≤ 4 · c(G) for planar graphs G +with ∆(G) ≤ 4. With c(G) ≤ 3 for all planar graphs G [1], item 2 in Theorem 1 follows. +3 +Robber wins sometimes if the maximum degree is at least five +In this section we prove item 3 in Theorem 1, i.e., that c∗(G) = Ω +�� +log(n) +� +for some +n-vertex plane graphs G with ∆(G) ≥ 5. We utilize a result of Nisse and Suchan [9] about +the cop number cp,q(G) for a different variant of Cops and Robber for any graph G and + +4 +Primal-Dual Cops and Robber +Figure 3 G4,2,2: An n × n grid with each edge subdivided four times and two rings. Faces are +colored according to their closest grid vertex. Deep and shallow faces are light and dark, respectively. +positive integers p and q. Here (as in the classical variant) the cops and the robber are on +the vertices of G. However, in each turn the cops may traverse up to p edges of G, while the +robber may traverse up to q edges of G. We refer to p and q as the velocities of the cops and +the robber, respectively. +▶ Theorem 4 ([8, 9]). Let Gn be the n × n grid graph, p be the velocity of the cops and q be +the velocity of the robber. If p < q, then cp,q(Gn) = Ω +�� +log(n) +� +. +The idea to prove item 3 in Theorem 1 is to construct a “grid-like” graph Gn,s,r for +positive integers n, s, r in which the robber in the primal-dual variant can move around faster +than the cops. Then she can simulate the evasion strategy of the robber in the variant of +Nisse and Suchan. +We start with the n × n grid graph Gn, n ≥ 3, with a planar embedding such that the +4-faces are the inner faces. We call the vertices of Gn the grid vertices. Then, each edge +of Gn is subdivided by 2s new vertices, called subdivision vertices, to obtain Gn,s. Two grid +vertices are called neighboring if they are adjacent in Gn. Further, inside each inner face of +Gn,s we add r nested cycles, called rings, of length 12s each and call their vertices the ring +vertices. Between any two consecutive rings we add a planar matching of 12s edges. Each +inner face of Gn,s has 8s subdivision vertices on its boundary and 12s ring vertices on its +outermost ring. At last, we add (in a crossing-free way) three edges from each subdivision +vertex to the outermost ring vertices in the two incident faces of Gn,s such that two edges +go to one ring, the third edge to the other ring, and every ring vertex receives exactly one +such edge. Along the 2s vertices of each subdivision path in Gn,s the side with two edges to +the ring should always switch. Thus each inner face of Gn,s receives 12s edges which are +connected to the 12s vertices of the outermost ring such that the drawing remains planar. +Call the resulting graph Gn,s,r and note that ∆(Gn,s,r) = 5. See also Figure 3. We +shall use a robber strategy in which she only focuses on grid vertices and moves between +these through the paths of subdivision vertices, i.e., only plays on Gn,s. The purpose of the +additional rings in Gn,s,r is to slow down the cops and force them to stay close to grid and +subdivision vertices, too, thereby simulating the game of Nisse and Suchan on Gn. +Formally, we call an inner face of Gn,s,r shallow if it is incident to some subdivision +vertex, and deep otherwise. Our first lemma implies that, due to the number of rings, cops +should not use deep faces. +▶ Lemma 5. Let a1, a2 be two shallow faces of Gn,s,r inside the same inner face A of Gn. + +M. T. Ha, P. Jungeblut and T. Ueckerdt +5 +If r > 3s, then any cop moving from a1 to a2 along a shortest path without leaving A uses +only shallow faces. +Proof of Lemma 5. First observe that there are exactly 12s shallow faces inside A; one for +each edge of the outermost ring. Hence, the cop may move from a1 to a2 using only shallow +faces in no more than 6s steps. On the other hand, the deep face b inside the innermost +ring is at distance r > 3s from each of a1, a2 and hence no shortest path between a1 and a2 +uses b. +Let H be the subgraph of the plane dual of Gn,s,r induced by all inner faces inside A, +except b. Then H ∼= Pr □ C12s is a square grid on a cylinder of height r and circumference 12s, +with the shallow faces forming a boundary cycle C. Since a1, a2 are on C and each shortest +path lies inside H, such path is contained in C, i.e., uses only shallow faces. +◀ +We have to hinder the cops from taking shortcuts through the outer face f0 of Gn,s,r. To +this end let G′ +n,s,r be a copy of Gn,s,r with outer face f ′ +0. Change the outer face of G′ +n,s,r +such that f ′ +0 is an inner face (while not changing the cyclic ordering of the edges around the +vertices) and define Gn,s,r to be the graph obtained from gluing Gn,s,r into face f ′ +0 of G′ +n,s,r +and identifying corresponding vertices. The robber will always stay on vertices of Gn,s,r and +whenever a cop uses a vertex v′ of G′ +n,s,r she acts as if he was on the corresponding vertex v +of Gn,s,r. Without loss of generality, we can therefore assume below that the game is played +on Gn,s,r with the cops being prohibited to enter the outer face. +For a face f ∈ F, we denote by vf be the grid vertex closest to f, breaking ties arbitrarily. +▶ Lemma 6. Let a, b be two shallow faces whose closest grid vertices va, vb have distance d +in Gn. If r > 3s, then in Gn,s,r the robber moving from va to vb needs at most (2s + 1)d +steps, while any cop moving from a to b needs at least 3s(d − 4) steps. +Proof of Lemma 6. For the first part it is enough to observe that the robber may go along +subdivision vertices, taking exactly 2s + 1 steps for every corresponding edge in Gn. +For the second part, i.e., the lower bound on the number of moves for a cop, let A +and B be the inner faces of Gn containing the inner faces a and b of Gn,s,r, respectively. +We assume that d ≥ 5, as otherwise 3s(d − 4) ≤ 0 and there is nothing to show, and hence +we have A ̸= B. More precisely, traveling from a to b, the cop must traverse (inner faces +of Gn,s,r corresponding to) at least d − 1 different inner faces of Gn. Cutting off the initial +part inside A and final part inside B, Lemma 5 implies that the remaining shortest path for +the cop uses only shallow faces. Thus, on her way, the cop visits shallow faces incident to at +least d − 3 distinct grid vertices, i.e., d − 4 transitions from a shallow face at a grid vertex to +a shallow face at a neighboring grid vertex. As each such transition requires 3s moves, the +claim follows. +◀ +Proof of item 3 in Theorem 1. Nisse and Suchan [9] (see also [8] for the omitted proofs) +describe an evasion strategy for a robber with velocity q that requires Ω +�� +log(n) +� +vertex-cops +with velocity p to capture him in Gn, provided q > p; see Theorem 4. We describe how +a robber with velocity 1 in Gn,s,r (for sufficiently large n, s, r) can simulate this strategy +against face-cops with velocity 1. +We choose p = 15, q = 16 and consider the game of Nisse and Suchan for these velocities. +For their graph Gn in which the robber can win against k = Ω +�� +log(n) +� +vertex-cops, we +then consider Gn,s,r with s = 16 and r = 3s + 1 = 49. Now we copy the evasion strategy S +for the robber as follows: Whenever it is the robber’s turn and the face-cops occupy faces +f1, f2, . . . , fk in Gn,s,r, consider the corresponding situation in Gn where the vertex-cops +occupy vf1, vf2, . . . , vfk. Based on these positions, S tells the robber to go to a vertex v at + +6 +Primal-Dual Cops and Robber +distance d ≤ q = 16 from the current position of the robber in Gn. By Lemma 5, the robber +in Gn,r,s can go to v in at most (2s + 1)d ≤ (2 · 16 + 1) · 16 = 528 turns. +In the meantime, each face-cop also makes up to 528 moves in Gn,r,s, traveling from some +face a to some face b, which is interpreted in Gn as the corresponding vertex-cop traveling +from va to vb. For va and vb to be at distance d′ ≥ 16 in Gn, by Lemma 5 the face-cop needs +at least 3s(d′ − 4) ≥ 3 · 16 · 12 = 576 turns, which is strictly more than 528. Thus, after 528 +turns, each vertex-cop made at most p = 15 steps in Gn, as required for strategy S. +Hence, the robber can evade k face-cops in Gn,s,r, proving c(Gn,s,r) > k. Since Gn,s,r +for s, r ∈ O(1) has O(n2) vertices, this completes the proof. +◀ +4 +Conclusions +Let c∗ +∆ denote the largest primal-dual cop number among all plane graphs with maximum +degree ∆. We have shown that c∗ +3 = 3, c∗ +4 ≤ 12 (this bound is certainly not optimal), and +c∗ +5 = ∞, while it is easy to see that c∗ +1 = 1, c∗ +2 = 2, and c∗ +∆ = ∞ for all ∆ > 5. Let us remark +that our proof for ∆ = 5 also holds for a variant of the game where the robber is already +captured when one cop is on one incident face. On the other hand, our proof for ∆ = 3 holds +verbatim to prove that three cops also suffice in a variant of the game where the graph is +embedded without crossings in any other surface, which makes it is interesting to consider +∆ = 4 here. +References +1 +Martin S. Aigner and M. Fromme. A Game of Cops and Robbers. Discrete Applied Mathematics, +8(1):1–12, 1984. doi:10.1016/0166-218X(84)90073-8. +2 +Anthony Bonato. An Invitation to Pursuit-Evasion Games and Graph Theory. American +Mathematical Society, 2022. +3 +Anthony Bonato and Richard J. Nowakowski. The Game of Cops and Robbers on Graphs. +American Mathematical Society, 2011. doi:10.1090/stml/061. +4 +Peter Bradshaw and Seyyed Aliasghar Hosseini. Surrounding Cops and Robbers on Graphs of +Bounded Genus, 2019. arXiv:1909.09916. +5 +Andrea C. Burgess, Rosalind A. Cameron, Nancy E. Clarke, Peter Danziger, Stephen Finbow, +Caleb W. Jones, and David A. Pike. Cops that surround a robber. Discrete Applied Mathematics, +285:552–566, 2020. doi:10.1016/j.dam.2020.06.019. +6 +Danny Crytser, Natasha Komarov, and John Mackey. Containment: A Variation of Cops and +Robber. Graphs and Combinatorics, 36(3):591–605, 2020. doi:10.1007/s00373-020-02140-5. +7 +Andrzej Dudek, Przemysław Gordinowicz, and Paweł Prałat. Cops and Robbers playing on +edges. Journal of Combinatorics, 5(1):131–153, 2014. doi:10.4310/JOC.2014.v5.n1.a6. +8 +Fedor V. Fomin, Petr A. Golovach, Jan Kratochvíl, Nicolas Nisse, and Karol Suchan. Pursuing +a fast robber on a graph. Theoretical Computer Science, 411(7–9):1167–1181, 2010. doi: +10.1016/j.tcs.2009.12.010. +9 +Nicolas Nisse and Karol Suchan. Fast Robber in Planar Graphs. In Hajo Broersma, Thomas +Erlebach, Tom Friedetzky, and Daniel Paulusma, editors, Graph-Theoretic Concepts in Com- +puter Science (WG 2008), volume 5344 of Lecture Notes in Computer Science, pages 312–323, +2008. doi:10.1007/978-3-540-92248-3_28. +10 +Richard J. Nowakowski and Peter Winkler. Vertex-to-Vertex Pursuit in a Graph. Discrete +Mathematics, 43(2–3):235–239, 1983. doi:10.1016/0012-365X(83)90160-7. +11 +Paweł Prałat. Containment Game Played on Random Graphs: Another Zig-Zag Theorem. +The Electronic Journal of Combinatorics, 22(2), 2015. doi:10.37236/4777. +12 +Alain Quilliot. Jeux et pointes fixes sur les graphes. PhD thesis, Université de Paris VI, 1978. + diff --git a/8dE5T4oBgHgl3EQfQg7H/content/tmp_files/load_file.txt b/8dE5T4oBgHgl3EQfQg7H/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d6e20584912f186fd43e9d749ef7400116decf6f --- /dev/null +++ b/8dE5T4oBgHgl3EQfQg7H/content/tmp_files/load_file.txt @@ -0,0 +1,281 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf,len=280 +page_content='Primal-Dual Cops and Robber Minh Tuan Ha � Karlsruhe Institute of Technology, Germany Paul Jungeblut � Karlsruhe Institute of Technology, Germany Torsten Ueckerdt � Karlsruhe Institute of Technology, Germany Abstract Cops and Robber is a family of two-player games played on graphs in which one player controls a number of cops and the other player controls a robber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In alternating turns, each player moves (all) his/her figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The cops try to capture the robber while the latter tries to flee indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In this paper we consider a variant of the game played on a planar graph where the robber moves between adjacent vertices while the cops move between adjacent faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The cops capture the robber if they occupy all incident faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We prove that a constant number of cops suffices to capture the robber on any planar graph of maximum degree ∆ if and only if ∆ ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 2012 ACM Subject Classification Mathematics of computing → Discrete mathematics → Combi- natorics → Combinatoric problems Keywords and phrases Cops and robber, planar graph, dual graph 1 Introduction Cops and Robber is probably the most classical combinatorial pursuit-evasion game on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The robber models an intruder in a network that the cops try to capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Two players play with complete information on a fixed finite graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The cop player controls a set of k cops, each occupying a vertex of G (possibly several cops on the same vertex), while the robber player controls a single robber that also occupies a vertex of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The players take alternating turns, where the cop player in his turn can decide for each cop individually whether to stay at its position or move the cop along an edge of G onto an adjacent vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Similarly, the robber player on her turn can leave the robber at its position or move it along an edge of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The cop player starts by choosing starting positions for his k cops and wins the game as soon as at least one cop occupies the same vertex as the robber, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=', when the robber is captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The robber player, seeing the cops positions, chooses the starting position for her robber and wins if she can avoid capture indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The least integer k for which, assuming perfect play on either side, k cops can always capture the robber, is called the cop number of G, usually denoted by c(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In this paper, we introduce Primal-Dual Cops and Robber which is played on a plane graph G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=', with a fixed plane embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Here, the cops occupy the faces of G and can move between adjacent faces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=', faces that share an edge), while the robber still moves along edges between adjacent vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In this game, the robber is captured if every face incident to the robber’s vertex is occupied by at least one cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Analogously, we call the least integer k for which k cops can always capture the robber in the Primal-Dual Cops and Robber game the primal-dual cop number of G and denote it by c∗(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' An obvious lower bound for c∗(G) is the maximum number of faces incident to any vertex in G: The robber can choose such a vertex as its start position and just stay there indefinitely (note that there is no zugzwang, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=', no obligation to move during ones turn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In particular, if G has maximum degree ∆(G) and there exists a vertex v with deg(v) = ∆(G), which is not a cut-vertex, then c∗(G) ≥ ∆(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=', c∗(K2,n) = ∆(K2,n) = n for any n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='05514v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='CO] 13 Jan 2023 2 Primal-Dual Cops and Robber Our contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We investigate, whether the primal-dual cop number c∗(G) is bounded in terms of ∆(G) for all plane graphs G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The answer is ‘Yes’ if ∆(G) ≤ 4 and ‘No’ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' ▶ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Each of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' For every plane graph G with ∆(G) ≤ 3 we have c∗(G) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' For every plane graph G with ∆(G) ≤ 4 we have c∗(G) ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' For some n-vertex plane graphs G with ∆(G) = 5 we have c∗(G) = Ω �� log(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Let us just briefly mention that Cops and Robber was introduced by Nowakowski and Winkler [10] and Quillot [12] for one cop and Aigner and Fromme [1] for k cops 40 years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Since then numerous results and variants were presented, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=', [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Perhaps most similar to our new variant are the recent surrounding variant of Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' [5] with vertex-cops and the containment variant of Cryster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' [6, 11] with edge-cops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In these variants the robber is captured if every adjacent vertex, respectively every incident edge, is occupied by a cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The smallest number of cops that always suffices for any planar graph G is 3 in the classical variant [1], 7 in the surrounding variant [4], 7∆(G) in the containment variant [6] and 3 when both, cops and robber, move on edges [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 2 Cops win always if the maximum degree is at most four We start with an observation that simplifies the proofs of items 1 and 2 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' ▶ Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Let the robber be on a vertex u with a neighbor v of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Then the robber is never required to move to v to evade the cops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' This is true because the set of faces required to capture the robber at v is a subset of the faces required to capture him at u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Further, his only possible moves at v are either staying there or moving back to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' As there is no zugzwang, he could just stay at u all along.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In both of the following proofs we assume that the graph contains only degree-3-vertices (respectively degree-4-vertices) and degree-1-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' This can always be achieved by adding leaves to vertices not yet having the correct degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Proof of item 1 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We give a winning strategy for three cops c1, c2, c3 in a planar graph G with ∆(G) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' First the cops choose arbitrary faces to start on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Then the robber chooses its start vertex u, which we assume to be of degree 3 by Observation 2 (it is trivial to capture him if all vertices have degree 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Let ∠u 1, ∠u 2, ∠u 3 be the three angles incident to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We denote the face containing an angle ∠ by f(∠) and define for each cop ci a target face fi, i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Initially we set fi = f(∠u i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The goal of each cop is to reach his target face, thereby capturing the robber when all three cops arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' If the robber moves, each cop updates his target face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Our strategy guarantees that the total distance of all three cops to their targets faces decreases over time, so it reaches zero after finitely many turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Clearly, in every game the robber has to move at some point to avoid being captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Assume that the robber moves from vertex u to vertex v (both of degree 3 by Observation 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Without loss of generality the angles around u and v are labeled as in Figure 1 with fi = f(∠u i ) being the current target face of cop ci, i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' First assume that c3 (or symmetrically c2) has not reached his target face yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In this case we assign the new target faces f1 = f(∠v 1), f2 = f(∠v 2) and f3 = f(∠v 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Note that for i = 1, 2 faces f(∠u i ) and f(∠v i ) are adjacent, so cop ci can keep his distance to his target face unchanged (or even decrease it) during his next turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Further note that f(∠u 3) = f(∠v 3), M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Ha, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Jungeblut and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Ueckerdt 3 ̸ u 1 ̸ u 2 ̸ u 3 ̸ v 1 ̸ v 2 ̸ v 3 u v w Figure 1 Labeling of the angles for a robber move from u to v (and possibly further to w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' v u ̸ u 1 ̸ u 2 ̸ u 3 ̸ v 1 ̸ v 2 ̸ v 3 ̸ u 4 ̸ v 4 Figure 2 A vertex cop and its four accompanying face-cops moving from u to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' so cop c3 can even decrease his distance by one during the next turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Thus the total distance of the three cops to their target faces decreased by at least one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' It remains the case that c2 and c3 have already reached their target faces (but c1 did not, as the game would be over otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In this case we move c1 one step towards his target face f1 = f(∠u 1) and c2, c3 both to f(∠v 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Now its the robber’s turn again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' If she does not move, we assign target faces fi = f(∠v i ), i = 1, 2, 3, and the total distance decreases after the cops’ next turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' If she moves back to u, we assign target faces fi = f(∠u i ), i = 1, 2, 3, and the total distance decreases after the cops’ next turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The last possibility for the robber is to move towards another neighbor w of v, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Then we assign f1 = f(∠v 1) and f2, f3 to be the faces containing the other two angles at w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In their next turn, c2 and c3 can again reach their target faces, while c1 can decrease his distance to his target face f(∠v 1) by one compared to the initial situation with the robber at vertex u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Again, the total distance is decreased, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' ◀ To prove item 2 in Theorem 1, we reduce our Primal-Dual Cops and Robber to the classical Cops and Robber with cops on vertices of G and then use a result from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' ▶ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In a plane graph G with ∆(G) ≤ 4, four face-cops can simulate a vertex-cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Let c be a vertex-cop starting at a vertex u ∈ V (G) with up to four incident angles ∠u i (for i ∈ {1, 2, 3, 4}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We place four face-cops on the (up to) four faces f(∠u i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' If the vertex-cop moves to an adjacent vertex v, the four face cops around it can in one step also move to faces containing the angles incident to v, see Figure 2 for the case that u and v both have degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' For vertices of degree less then 4 it only gets easier for the face-cops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' ◀ An immediate corollary of Lemma 3 is that c∗(G) ≤ 4 · c(G) for planar graphs G with ∆(G) ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' With c(G) ≤ 3 for all planar graphs G [1], item 2 in Theorem 1 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 3 Robber wins sometimes if the maximum degree is at least five In this section we prove item 3 in Theorem 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=', that c∗(G) = Ω �� log(n) � for some n-vertex plane graphs G with ∆(G) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We utilize a result of Nisse and Suchan [9] about the cop number cp,q(G) for a different variant of Cops and Robber for any graph G and 4 Primal-Dual Cops and Robber Figure 3 G4,2,2: An n × n grid with each edge subdivided four times and two rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Faces are colored according to their closest grid vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Deep and shallow faces are light and dark, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' positive integers p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Here (as in the classical variant) the cops and the robber are on the vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' However, in each turn the cops may traverse up to p edges of G, while the robber may traverse up to q edges of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We refer to p and q as the velocities of the cops and the robber, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' ▶ Theorem 4 ([8, 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Let Gn be the n × n grid graph, p be the velocity of the cops and q be the velocity of the robber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' If p < q, then cp,q(Gn) = Ω �� log(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The idea to prove item 3 in Theorem 1 is to construct a “grid-like” graph Gn,s,r for positive integers n, s, r in which the robber in the primal-dual variant can move around faster than the cops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Then she can simulate the evasion strategy of the robber in the variant of Nisse and Suchan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We start with the n × n grid graph Gn, n ≥ 3, with a planar embedding such that the 4-faces are the inner faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We call the vertices of Gn the grid vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Then, each edge of Gn is subdivided by 2s new vertices, called subdivision vertices, to obtain Gn,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Two grid vertices are called neighboring if they are adjacent in Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Further, inside each inner face of Gn,s we add r nested cycles, called rings, of length 12s each and call their vertices the ring vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Between any two consecutive rings we add a planar matching of 12s edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Each inner face of Gn,s has 8s subdivision vertices on its boundary and 12s ring vertices on its outermost ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' At last, we add (in a crossing-free way) three edges from each subdivision vertex to the outermost ring vertices in the two incident faces of Gn,s such that two edges go to one ring, the third edge to the other ring, and every ring vertex receives exactly one such edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Along the 2s vertices of each subdivision path in Gn,s the side with two edges to the ring should always switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Thus each inner face of Gn,s receives 12s edges which are connected to the 12s vertices of the outermost ring such that the drawing remains planar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Call the resulting graph Gn,s,r and note that ∆(Gn,s,r) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' See also Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We shall use a robber strategy in which she only focuses on grid vertices and moves between these through the paths of subdivision vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=', only plays on Gn,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The purpose of the additional rings in Gn,s,r is to slow down the cops and force them to stay close to grid and subdivision vertices, too, thereby simulating the game of Nisse and Suchan on Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Formally, we call an inner face of Gn,s,r shallow if it is incident to some subdivision vertex, and deep otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Our first lemma implies that, due to the number of rings, cops should not use deep faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Let a1, a2 be two shallow faces of Gn,s,r inside the same inner face A of Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Ha, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Jungeblut and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Ueckerdt 5 If r > 3s, then any cop moving from a1 to a2 along a shortest path without leaving A uses only shallow faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' First observe that there are exactly 12s shallow faces inside A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' one for each edge of the outermost ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Hence, the cop may move from a1 to a2 using only shallow faces in no more than 6s steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' On the other hand, the deep face b inside the innermost ring is at distance r > 3s from each of a1, a2 and hence no shortest path between a1 and a2 uses b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Let H be the subgraph of the plane dual of Gn,s,r induced by all inner faces inside A, except b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Then H ∼= Pr □ C12s is a square grid on a cylinder of height r and circumference 12s, with the shallow faces forming a boundary cycle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Since a1, a2 are on C and each shortest path lies inside H, such path is contained in C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=', uses only shallow faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' ◀ We have to hinder the cops from taking shortcuts through the outer face f0 of Gn,s,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' To this end let G′ n,s,r be a copy of Gn,s,r with outer face f ′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Change the outer face of G′ n,s,r such that f ′ 0 is an inner face (while not changing the cyclic ordering of the edges around the vertices) and define Gn,s,r to be the graph obtained from gluing Gn,s,r into face f ′ 0 of G′ n,s,r and identifying corresponding vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The robber will always stay on vertices of Gn,s,r and whenever a cop uses a vertex v′ of G′ n,s,r she acts as if he was on the corresponding vertex v of Gn,s,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Without loss of generality, we can therefore assume below that the game is played on Gn,s,r with the cops being prohibited to enter the outer face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' For a face f ∈ F, we denote by vf be the grid vertex closest to f, breaking ties arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' ▶ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Let a, b be two shallow faces whose closest grid vertices va, vb have distance d in Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' If r > 3s, then in Gn,s,r the robber moving from va to vb needs at most (2s + 1)d steps, while any cop moving from a to b needs at least 3s(d − 4) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' For the first part it is enough to observe that the robber may go along subdivision vertices, taking exactly 2s + 1 steps for every corresponding edge in Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' For the second part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=', the lower bound on the number of moves for a cop, let A and B be the inner faces of Gn containing the inner faces a and b of Gn,s,r, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We assume that d ≥ 5, as otherwise 3s(d − 4) ≤ 0 and there is nothing to show, and hence we have A ̸= B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' More precisely, traveling from a to b, the cop must traverse (inner faces of Gn,s,r corresponding to) at least d − 1 different inner faces of Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Cutting off the initial part inside A and final part inside B, Lemma 5 implies that the remaining shortest path for the cop uses only shallow faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Thus, on her way, the cop visits shallow faces incident to at least d − 3 distinct grid vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=', d − 4 transitions from a shallow face at a grid vertex to a shallow face at a neighboring grid vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' As each such transition requires 3s moves, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' ◀ Proof of item 3 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Nisse and Suchan [9] (see also [8] for the omitted proofs) describe an evasion strategy for a robber with velocity q that requires Ω �� log(n) � vertex-cops with velocity p to capture him in Gn, provided q > p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We describe how a robber with velocity 1 in Gn,s,r (for sufficiently large n, s, r) can simulate this strategy against face-cops with velocity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We choose p = 15, q = 16 and consider the game of Nisse and Suchan for these velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' For their graph Gn in which the robber can win against k = Ω �� log(n) � vertex-cops, we then consider Gn,s,r with s = 16 and r = 3s + 1 = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Now we copy the evasion strategy S for the robber as follows: Whenever it is the robber’s turn and the face-cops occupy faces f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' , fk in Gn,s,r, consider the corresponding situation in Gn where the vertex-cops occupy vf1, vf2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' , vfk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Based on these positions, S tells the robber to go to a vertex v at 6 Primal-Dual Cops and Robber distance d ≤ q = 16 from the current position of the robber in Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' By Lemma 5, the robber in Gn,r,s can go to v in at most (2s + 1)d ≤ (2 · 16 + 1) · 16 = 528 turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In the meantime, each face-cop also makes up to 528 moves in Gn,r,s, traveling from some face a to some face b, which is interpreted in Gn as the corresponding vertex-cop traveling from va to vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' For va and vb to be at distance d′ ≥ 16 in Gn, by Lemma 5 the face-cop needs at least 3s(d′ − 4) ≥ 3 · 16 · 12 = 576 turns, which is strictly more than 528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Thus, after 528 turns, each vertex-cop made at most p = 15 steps in Gn, as required for strategy S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Hence, the robber can evade k face-cops in Gn,s,r, proving c(Gn,s,r) > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Since Gn,s,r for s, r ∈ O(1) has O(n2) vertices, this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' ◀ 4 Conclusions Let c∗ ∆ denote the largest primal-dual cop number among all plane graphs with maximum degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' We have shown that c∗ 3 = 3, c∗ 4 ≤ 12 (this bound is certainly not optimal), and c∗ 5 = ∞, while it is easy to see that c∗ 1 = 1, c∗ 2 = 2, and c∗ ∆ = ∞ for all ∆ > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Let us remark that our proof for ∆ = 5 also holds for a variant of the game where the robber is already captured when one cop is on one incident face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' On the other hand, our proof for ∆ = 3 holds verbatim to prove that three cops also suffice in a variant of the game where the graph is embedded without crossings in any other surface, which makes it is interesting to consider ∆ = 4 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' References 1 Martin S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Aigner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Fromme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' A Game of Cops and Robbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Discrete Applied Mathematics, 8(1):1–12, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='1016/0166-218X(84)90073-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 2 Anthony Bonato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' An Invitation to Pursuit-Evasion Games and Graph Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' American Mathematical Society, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 3 Anthony Bonato and Richard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Nowakowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The Game of Cops and Robbers on Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' American Mathematical Society, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='1090/stml/061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 4 Peter Bradshaw and Seyyed Aliasghar Hosseini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Surrounding Cops and Robbers on Graphs of Bounded Genus, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='09916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 5 Andrea C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Burgess, Rosalind A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Cameron, Nancy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Clarke, Peter Danziger, Stephen Finbow, Caleb W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Jones, and David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Pike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Cops that surround a robber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Discrete Applied Mathematics, 285:552–566, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 6 Danny Crytser, Natasha Komarov, and John Mackey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Containment: A Variation of Cops and Robber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Graphs and Combinatorics, 36(3):591–605, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='1007/s00373-020-02140-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 7 Andrzej Dudek, Przemysław Gordinowicz, and Paweł Prałat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Cops and Robbers playing on edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Journal of Combinatorics, 5(1):131–153, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='4310/JOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 8 Fedor V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Fomin, Petr A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Golovach, Jan Kratochvíl, Nicolas Nisse, and Karol Suchan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Pursuing a fast robber on a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Theoretical Computer Science, 411(7–9):1167–1181, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 9 Nicolas Nisse and Karol Suchan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Fast Robber in Planar Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' In Hajo Broersma, Thomas Erlebach, Tom Friedetzky, and Daniel Paulusma, editors, Graph-Theoretic Concepts in Com- puter Science (WG 2008), volume 5344 of Lecture Notes in Computer Science, pages 312–323, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='1007/978-3-540-92248-3_28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 10 Richard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Nowakowski and Peter Winkler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Vertex-to-Vertex Pursuit in a Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Discrete Mathematics, 43(2–3):235–239, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='1016/0012-365X(83)90160-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 11 Paweł Prałat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Containment Game Played on Random Graphs: Another Zig-Zag Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' The Electronic Journal of Combinatorics, 22(2), 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content='37236/4777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' 12 Alain Quilliot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' Jeux et pointes fixes sur les graphes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} +page_content=' PhD thesis, Université de Paris VI, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'} diff --git a/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf b/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..679e8ba6fc50af71a18342b8ce84b1404b05fa68 --- /dev/null +++ b/99AyT4oBgHgl3EQfRPYW/content/2301.00060v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6230f7c04d72c3aa6e5cdad131b13461e2a2f574013259b955154e64117e8eb3 +size 6124651 diff --git a/99AyT4oBgHgl3EQfRPYW/vector_store/index.faiss b/99AyT4oBgHgl3EQfRPYW/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..fb53f16464568b9e0961cd1ee27e4bb21addeba6 --- /dev/null +++ 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The purpose of this paper is to utilize statistical methodologies to infer from market +prices of assets and their derivatives the magnitude of the set of a measure M that defines acceptance +sets of risky future cash flows. We assume that M contains the collection of bilateral gamma random +variables, and estimate upper and lower boundaries of the compensation needed for a given bilateral +gamma distributed future cash flow to be acceptable. We show that prospects theory provides a +natural interpretation of the behaviors implied by such boundaries, which are not compatible with +expected utility theory. Boundaries for bilateral gamma risk neutral scale parameters for given speed +parameters are also estimated and tested against market data and, in particular, comparisons are +made with known empirical facts about the magnitude of the acceptance set of a common class of +risk measures. +1. Introduction +The definition of acceptable risks, based on the axiomatization of the concept of coherent risk +measure given in Artzner et al. (1999) and their convex generalization (Follmer & Schied (2002)), +is a major recent advance in mathematical finance, as, among other applications, it provides an +operative framework for superhedging in incomplete markets. Starting from a monetary measure, +such as Value at Risk, that only satisfies the basic requirements of monotonicity and cash invariance, +practical considerations (e.g. that the combined exposure of two trading desks ought to be less +risky than that of the two desks taken separately, or that lack of liquidity may affect the future net +worth of a single, large, position) lead one to require that a measure of risk also satisfy subadditivity +and positive homogeneity. A measure of risk ρ then defines a set Aρ of acceptable risks as those +random variables X such that ρ(X) ≥ 0. Conversely, it is possible to show that given a cone A of +acceptable risks, the functional +ρ(X) = inf{m ∈ R : m + X ∈ A} +(1.1) +satisfies monotonicity, cash invariance, subadditivity and positive homogeneity. Based on convex +duality, a risk measure is also specified by a set of equivalent probability measures M as +ρ(X) = inf +Q∈M EQ[X]. +(1.2) +The class M can be interpreted as the set of possible and credible macroeconomic/financial models, +so that 1.2 is referred to as the robust representation of ρ, and risk measures become natural tools +for the purpose of modeling uncertainty. For convex risk measures, a penalty α(Q) is added to 1.2 +to take into account that some models Q ∈ M may be more or less plausible than others. +E-mail address: yshirai@umd.edu. +Date: January 16, 2023. +2020 Mathematics Subject Classification. 60G18, 60G51, 91G20. +Key words and phrases. Bilateral Gamma, Prospects Theory, Knightian Uncertainty, Risk Measures, Nonlinear +Levy Processes, Diffusion Map, Quantile Regression, Distorted Regression, Gaussian Process Regression. +1 +arXiv:2301.05333v1 [q-fin.MF] 13 Jan 2023 + +2 +ACCEPTABLE BILATERAL GAMMA PARAMETERS +Typical examples of risk measures are those based on certainty equivalent, such as the entropic +risk measure, which are known in general as utility-based shortfall risk measures and are defined +by the acceptance set +A = {X : E[u(X)] ≥ u(c)} +for a given convex utility u and a threshold c, and those obtained by modifying the tails of the +underline statistical measure P, such as the expected shortfall, which are known in general as +spectral risk measures and are defined by the Choquet integral +ρ(X) = +� ∞ +0 +Ψ(P(X+ ≥ a))da − +� ∞ +0 +ˆΨ(P(X− ≥ a))da, +where Ψ : [0, 1] → [0, 1] is increasing and convex and ˆΨ(u) = 1 − Ψ(1 − u). +As the above examples confirm, relatively little is known in general about the set M. Note, +however, that a risk measure is an expected value under a worst case scenario measure, and, as +such, it defines a minimal current valuation (or maximal bid price) of the future cash flow X, while +−ρ(−X) gives a maximal valuation (or the minimal ask price). Assuming that market prices of +traded assets are random variables whose distribution belong to a specific class and is determined by +a set Θ ∈ RD of parameters, observed market prices imply specific boundaries for the set Θ and, in +turn, for M. For instance, if M is (or contains) the class of normal random variables parameterized +by pairs (µ, σ2) of mean and variance of assets returns, one can ask what are maximal and minimal +bounds for µ given σ2 that are implied by historically observed pairs (µ, σ2) of traded assets, in +turn estimated from market prices. These bounds are then naturally interpreted as structural limits +for the reward µ given the risk σ2 that the economic system can offer without compromising its +financial stability, as defined by the regulator. +To fix a reference framework, consider a market composed only of one risky asset with log return +X and a riskless one in zero net supply with zero risk free rate. Then, +1 = EQ[eX] = E[ηeX], +(1.3) +where Q is a risk neutral measure, η the corresponding stochastic discount factor. If the distribution +of X under the statistical measure P is parameterized by θ ∈ RD, and assuming the existence of +a representative investor with utility U defined by a set of parameters ξ ∈ Rm, there is a function +V : RD × Rm → R that evaluates to 1 at (θ, ξ). Specifically (see e.g. Madan (2020a)), the risk +neutral density (with respect to the log return) is given by +h(x, θ, ξ) = +U ′ +ξ(ex)fθ(x) +� +R U ′ +ξ(es)fθ(s)ds, +(1.4) +where fθ is the statistical density of X. Based on 1.3 and 1.4, if the prospects offered by the risky +asset suddenly deteriorate, 1with θ replaced by a riskier θ′, a decrease in the equilibrium risk free +rate is needed to compensate. In extreme cases, however, investors may no longer be allowed to +hold such an asset which will be liquidated and may, ultimately, stop trading in some markets. As +an example, one may think of pension funds, which are not allowed to hold speculative grade bonds, +or to those asset classes, such as hedge funds, that are only reserved to institutional investors. +In the case of normal returns, as it is well known (Markovitz (1952), Tobin (1958), Sharpe (1964), +Lintner (1965)), the efficient frontier essentially provides the upper limit for the reward µp given +a risk defined by σ2, and also the lower one, as this is the upper limit for a short position. In +general, however, this result lies on the assumption that investors have mean-variance preferences, +and that, in particular, they are expected utility maximizers. Empirical observations, on the other +hand, have shown in many occasions that asset returns are not compatible with such axioms - a +well known example being the equity premium puzzle, according to which U.S. equity risk premia + +ACCEPTABLE BILATERAL GAMMA PARAMETERS +3 +over Treasury Bills rates reflect an implausible level of aversion to risk under expected utility theory +(Mehra & Prescott (1985)). +An alternative to expected utility theory, termed “prospects theory”, is based on a series of ex- +periments conducted by psychologists D. Kahneman and A. Tverski (Kahneman & Tverski (1979)). +One of their results, in particular, is that humans tend to be risk seekers rather than risk averse in +the case of pure losses prospects. For instance, the prospect of winning 1000 dollars with probability +1/2 and winning zero otherwise is generally dominated by the prospect of winning 500 dollars with +probability 1, but the prospect of losing 1000 dollars with probability 1/2 and losing zero otherwise +dominates the prospect of losing 500 dollars with probability 1, independently of initial wealth. +Based on such evidence, one is then led to interpret an asset’s return as the sum of two prospects, +one consisting of pure gains, and the other one of pure losses, and investors rank different assets’ +returns based on the expectations and variances (µp, σ2 +p, µn, σ2 +n) of gains and losses. In particular, +higher variance of losses is compensated, ceteris paribus, by lower expectation µp of the gains. +The bilateral gamma distribution (Kuchler & Tappe (2008)) and its multivariate version (Madan +(2020b)) provide a natural modeling framework for such a preference specification for several rea- +sons. Firstly, it is the difference of two independent gamma variates, interpretable as gains and +losses, and it is completely specified by the vector (µp, σp, µn, σn) of their expected values and stan- +dard deviations. Secondly, even in a continuous time setting, the bilateral gamma process is the +difference of two independent gamma processes, while, for instance, path realizations of diffusion +processes have infinite variation. Thirdly, the bilateral gamma distribution provides a very good +fit to the (log) returns distribution implied by time series of returns and also by options prices +(Kuchler & Tappe (2008)), which shows that it is more suitable than, e.g., the normal distribution +for the purpose of modeling asset returns. Finally, as shown below, the expected utility of an asset +with bilateral gamma return X is a function F : (µp, σp, µn, σn) → E[u(X)], increasing in µp, and +decreasing in σp, µn and σn, so that under expected utility theory variations in (σp, µn, σn) are +compensated by variations of equal sign in µp. +Based on this considerations, we assume in this paper that the set of credible models M includes +the set of bilateral gamma random variables, and we learn bounds fM, fm : (σp, µp, σn) → µp for +µp given risks (σ2 +p, µn, σ2 +n) via quantile and/or distorted linear and/or Gaussian process regression. +An interesting result obtained is that both boundaries are generally increasing in (σp, µp), but +decreasing in σn, suggesting that investors, independently of their wealth, seek for lower (resp. +higher) risk when it comes to purely positive (resp. negative) processes. We test the boundaries +computed by assessing how well their implied performance measures (Sharpe ratio and acceptability +index) compare with those typically observed in the financial markets. Furthermore, we investigate +the linearity of fM and fm by comparing the results of a linear lower dimensional embedding and a +nonlinear one, and we show through a simple variation of a Lucas tree economy Lucas (1978) that +the behaviors observed are indeed consistent with prospects theory. +Finally, we move our attention to the risk neutral world, based on the suggestive interpretation +given in Madan (2020a) that, for bilateral gamma returns, the scale parameters (bp, bn) determine +the structure of limit orders, while the speed parameters (cp, cn) determine that of market orders. +It is then natural to assume that a relationship exists between the two pairs of parameters, in the +sense that for given (cp, cn), the scale parameters (bp, bn) are bounded to a specific range, as the +structure of market orders cannot be too independent from that of limit orders and viceversa. As +done for the statistical moments, the boundaries of such range are learned through quantile and +distorted regression. In this case, we determine theoretical boundaries as well based on the well +known robust representation of spectral risk measures (Madan & Schoutens (2021)), and evidence +is offered of their comparability with the empirically estimated ones. +The rest of the paper is organized as follows. First we show that for bilateral gamma returns, +risks and compensations are identified by the vector (σp, µn, σn) and µp respectively. Empirical + +4 +ACCEPTABLE BILATERAL GAMMA PARAMETERS +observations are reported in section 3, and the variation on Lucas Tree model is presented in +section 4. Risk neutral parameters are analyzed in 5. Section 6 concludes. +2. Bilateral Gamma Returns +2.1. From Brownian Motion to Bilateral Gamma Process. Given its central role in this +paper, the construction and properties of the bilateral gamma process are reviewed in this section. +In Black & Scholes (1973), F. Black and M. Scholes proposed to model the dynamics of log-returns as +a Brownian motion (GBM), as prices exhibit exponential growth and on the assumption, rooted in +an entropy maximization argument (Madan (2020a)), that log returns are asymptotically normally +distributed. +However, returns exhibit heavier tails than those implied by the normal distribution (Fama +(1965)) and frequent discontinuities in their path trajectories. In addition, risk aversion results in +periods of intense trading, determined by widespread selling in securities, alternating with lower +activity ones, thus implying that returns’ quadratic variation is not linear in time. It also results +in higher demand for out of the money (OTM) than for the corresponding OTM calls, generating +a volatility smile. +Another entropy maximization argument then suggests modeling economic time as a gamma +process, and stock market log returns as Brownian motion evaluated at such gamma time. The +resulting process, pioneered by D. Madan and E. Seneta (Madan & Seneta (1990)) and termed the +variance gamma process, is a pure jump Levy process with infinite activity and finite variation. In +fact, such process is the difference of two i.i.d. gamma processes, which naturally correspond to +gains and losses. Finally, motivated by the fact that downward jumps in prices are generally higher +than upward ones, the bilateral gamma process is defined as the difference of two independent +gamma processes with different shape and scale parameters (Kuchler & Tappe (2008)). The gains +and losses increments have BG distribution βΓ(bp, cp, bn, cn), defined by the convolution +βΓ(bp, cp, bn, cn) = Γ(bp, cp) ∗ Γ(−bn, cn), +where bp, cp, bn, cn > 0 and, for α > 0, λ ∈ R, a Γ(λ, α)-distributed random variable has density +f(x) = +1 +Γ(α)|λ|α |x|α−1e−|x|/|λ| � +11{λ>0}(x)11{x>0}(x) + 11{λ<0}(x)11{x<0}(x) +� +, x ∈ R +with Γ(α) the Gamma function at α. Then, expected value and standard deviation of gains and +losses, denoted respectively by µp, σp, µn and σn, are given by +µp = cpbp, σp = √cpbp, µn = cnbn, σn = √cnbn. +By the convolution theorem, the characteristic function of the increments in t units of time is +ϕt(u) = (1 − iubp)−tcp (1 + iubn)−tcn , +(2.1) +and it follows easily from 2.1 that BG densities are stable under convolution and are infinitely +divisible, and so the BG process is a well defined Levy process. From formula 2.1 and the Levy- +Khintchine representation we also deduce its Levy density to be +k(x) = +�cp +x e−x/bp11{(0,∞)}(x) + cn +|x|e−|x|/bn11{(−∞,0)}(x) +� +, x ∈ R +which shows that a BG process enjoys the self decomposability property.1 Then (see Carr et al. +(207) and the references therein) a BG distributed random variable X is a limit law, i.e. there +are centering and scaling constants {cn}n∈N and {bn}n∈N and a sequence {Zk}k∈N of i.i.d. random +variables such that the distribution of bnSn + cn converges in distribution to X, where Sn = +1A random variable X is self decomposable if for any 0 < c < 1 there is an independent random variable XC such +that X +d= cX + Xc. A Levy process enjoys the self decomposability property if its increments are self decomposable. + +ACCEPTABLE BILATERAL GAMMA PARAMETERS +5 +�n +k=1 Zk. This is a remarkable property, since if returns consist of some average of a large number +of independent news or other type of influences, it is reasonable to expect that their distribution +should be well approximated by a limit law. In the GBM case such law is the Gaussian, but, as +noticed in Carr et al. (207), there is “no compelling economic motivation” for the scaling constants +to be √n as in the classical central limit theorem. +Evidence of the goodness of fit of the BG density to returns distributions is presented in Kuchler +& Tappe (2008), where, using data on DAX between 1996 and 1998, it is shown that the null +hypothesis that the log returns distribution is in the BG class is not rejected. Furthermore, as +proved in Kuchler & Tappe (2008), for all BG parameters there exists a measure Q equivalent +to P such that, under Q, the discounted exponential BG process is a (local) martingale and an +exponential BG process, and one typically succeed in fitting the option prices surface, at least for +a single fixed maturity, through an exponential BG process. +2.2. Bilateral Gamma Returns under Expected Utility Theory. The notion and character- +izations of second order stochastic dominance (SSD) are recalled below (see Rothschild & Stiglitz +(1970)). +Definition 2.1. Given random variables X and Y , one says that X first (resp. second) order +stochastically dominates Y , i.e. X ⪰1 Y (resp. X ⪰2 Y ) if and only if E[u(X)] ≥ E[u(Y )] for +every increasing (resp. increasing and concave) real valued function u. +Theorem 2.2. Let X and Y be random variables with distribution functions F and G respectively. +Then, X ⪰1 Y if and only if G(t) ≥ F(t) for every t ∈ R. +Theorem 2.3. Let X and Y be random variables with distribution functions F and G respectively. +Then, the following are equivalent +(i) X ⪰2 Y ; +(ii) there are random variables Z and ε such that Y ∼ X + Z + ε, Z ≤ 0 and E[ε|X + Z] = 0; +(iii) +� t +−∞ G(s)ds ≥ +� s +−∞ F(s)ds for every t ∈ R. +In addition, if E[X] = E[Y ], then the following are equivalent: +(i) X ⪰2 Y ; +(ii) there is a random variable ε such that Y ∼ X + ε and E[ε|X + Z] = 0; +(iii) E[u(X)] ≥ E[u(Y )] for every u concave. +Corollary 2.4. Suppose X ⪰2 Y . Then, E[X] ≥ E[Y ] and if E[X] = E[Y ] then V (X) ≤ V (Y ). +Proof. That E[X] ≥ E[Y ] if X ⪰2 Y follows immediately from the fact that the identity is non +decreasing and concave. If E[X] = E[Y ], then E[u(X)] ≥ E[u(Y )] for every u concave, and so, +setting u(x) = −x2 + E[X], one obtains V (X) = E[X2 − E[X]] ≤ E[Y 2 − E[X]] = V [Y ]. +□ +Thus, for bilateral gamma returns, SSD implies higher expected gains and/or lower expected +losses, and, for equal expected gains and losses, lower standard deviation of gains and/or losses. A +partial converse of this statement is shown below, and is based on the following results. +Theorem 2.5. Let X and Y be random variables with densities f and g. If the likelihood ratio f +g +is monotonically increasing, than X ⪰1 Y . If the likelihood ratio is monotonically increasing on +(−∞, x0) ∪ (x1, ∞) and decreasing on (x0, x1), with x0 < x1 ∈ R, then X ⪰2 Y . +Proof. See Ali (1975) and the references therein. +□ +Theorem 2.6. Let X and Y be two gamma distributed random variable with scale and shape +parameters (b, c) and (b′, c′) respectively. Then, +(i) if b = b′, then c > c′ iff X ⪰2 Y ; +(ii) if c = c′, then b > b′ iff X ⪰2 Y ; + +6 +ACCEPTABLE BILATERAL GAMMA PARAMETERS +(ii) +c +c′ ≤ max(1, b′ +b ) with strict inequality at least when b′ +b = 1 iff X ⪰2 Y . +Proof. Based on showing that the assumptions of theorem 2.5 are satisfied. See Ali (1975). +□ +Corollary 2.7. Let X and Y be two gamma distributed random variables with scale and shape +parameters (b, c) and (b′, c′) respectively. +Then, X second order stochastically dominates Y if +E[X] ≥ E[Y ] and V [X] ≤ V [Y ] with at least one strict inequality. Similarly, −X second order +stochastically dominates −Y if E[X] ≤ E[Y ] and V [X] ≤ V [Y ] with at least one strict inequality. +Proof. Suppose E[X] ≥ E[Y ] and V [X] ≤ V [Y ] with at least one strict inequality, i.e. bc ≥ b′c′ and +b2c ≤ b′2c′ with at least one strict inequality. Then, c +c′ ≥ b′ +b , and +b′ +b = b′2c′ +b2c +bc +b′c′ > 1 +so X ⪰2 Y by Theorem 2.6. The result for −X and −Y follows from adapting Theorem 2.6 to the +case of the negative of gamma distributions. +□ +Corollary 2.8. Let X+, X−, Y +, Y − be four gamma distributed random variable with scale and +shape parameters (bp, cp), (bn, cn), (b′ +p, c′ +p) and (b′ +n, c′ +n) respectively. Then, X := X+ − X− second +order stochastically dominates Y := Y +−Y − if E[X+] ≥ E[Y +], E[X−] ≤ E[Y −], V [X+] ≤ V [Y +], +and V [X−] ≤ V [Y −] with exactly one strict inequality. +Proof. Suppose for instance E[X+] > E[Y +]. Then, by Corollary 2.7, X+ ⪰2 Y +, and so, for all +t ∈ [0, ∞) +� t +0 +F +(s) − G+(s)ds ≤ 0, +where F + and G+ denote the cumulative distribution function of X+ and Y + respectively. Then, +using Tonelli’s theorem, +� t +0 +F(s) − G(s)ds = +� ∞ +0 +� t +0 +F +(s − ξ) − G+(s − ξ)dsdF −(ξ) ≤ 0, +where F − is the (common) distribution of −X− and −Y −, and the conclusion follows from Theorem +2.3. The other cases are similar. +□ +Based on the last corollary and transitivity of SSD, the observation that a positive variation in +µp can compensate a positive variation in any among the upside volatility σp, the expected loss +prospect µn or the downside volatility σn is evidence of investors’ risk seeking behaviors. +Note that µp is not, in general, a “reward” accessible to an investor holding the asset. In fact, +for a given time horizon T the expected return for holding the asset is the value µ(T) that satisfies +S0eµ(T) = E[S0eXT ], and so the variation +lim +T↓0 +µ(T) +T += +� +R +(ex − 1)k(x)dx = (1 − bp)−cp(1 + bn)−cn +(2.2) +better serves this purpose. Thus, we refer to µp as a “compensation” for the risks (σp, µn, σn). +2.2.1. Log-Returns and Kelly’s Criterion. In the case log returns are assumed to be bilateral gamma +variates, these results cannot hold anymore, since, for instance, an increase in σp and/or σn im- +plies higher expected value of the return, and it cannot imply second order stochastic dominance. +However, a traditional assumption in the financial and economics literature, justified by some ev- +idence (Arrow (1971)), is to assume that investors maximize log-returns. In our context, such an +assumption implies that an asset is preferred to another one if and only if the expected log-return +is higher. More generally, for asset allocation problems, logarithmic utility yields the best return +in the long run, assuming the investor faces a long sequence of investment decisions (Kelly (1956), + +ACCEPTABLE BILATERAL GAMMA PARAMETERS +7 +Merton (1969), Cover (1991)), but for an investor with a short/medium term horizon, a logarithmic +utility will not capture aversion to short term high volatility (Samuelson (1979)), thus leading to +consider a utility specification with a coefficient of relative risk aversion (CRRA) bounded below by +1.2 It then follows from the results of this section and proposition 2.2.1 below that, for a reasonable +utility specification such as u(log(·)), risks and their compensation are captured by (σp, µn, σn) and +µp respectively even when log-returns belong to the bilateral gamma class. +Proposition 2.9. A strictly increasing and concave function v ∈ C2 ((0, ∞)) has CRRA coefficient +greater than 1 if and only if there is a strictly increasing and concave function u ∈ C2(R) such that +v(x) = u(log(x)) for every x ∈ (0, ∞). +Proof. Suppose such a u exists. Then, for all x ∈ (0, ∞), u′′(log(x)) ≥ 0 and u′(log(x)) < 0 +xv′′(x) +v′(x) = −x +d2 +dx2 u(log(x)) +d +dxu(log(x)) = 1 − u′′(log(x)) +u′(log(x)) ≥ 1. +On the other hand, if v has CRRA bounded below by 1, then, setting u(y) = v(ey) for every y ∈ R, +we obtain u′(y) = v′(ey)ey > 0 and u′′(y) = v′′(ey)e2y + v′(ey)ey ≤ 0. +□ +3. The Acceptance Set +3.1. Learning the Boundaries. As mentioned in the introduction, not all quadruples (µp, σp, µn, σn) +can be traded, or, in other words, there are structural limits to how high and/or low is the level +of rewards that can be offered for given risks. In order to determine such limits, moments of gains +and losses were estimated for 184 stocks (whose ticker is reported in appendix A) for the period +01/01/2008 to 31/12/2020 using one year of data for each estimate.3 Assuming the boundaries are +defined by functions fm, fM : (σp, µn, σn) → µp, we find fM and fm by solving, respectively, +min +f∈F(1 − τM) +� +i +[µp(i) − fM(σp(i), µn(i), σn(i))]+ − τM +� +i +[µp(i) − fM(σp(i), µn(i), σn(i))]− , +min +f∈F(1 − τm) +� +i +[µp(i) − fm(σp(i), µn(i), σn(i))]+ − τm +� +i +[µp(i) − fm(σp(i), µn(i), σn(i))]− , +where F is a suitable class of functions which is here assumed to be the class of linear Gaussian +process (GPR) regressors, τM = 0.95 and τm = 0.05. In our implementation of quantile GPR, +the kernel hyperparameters were estimated using the standard loss function, while the regression +coefficients are chosen to maximize the quantile loss function. Specifically, recall that GPR assumes +µp = α + h(σp, µnσn)T β + f(σp, µnσn) + ε, +(3.1) +where ε is noise with variance σ2 +ε, h is the map to features space (here assumed to be the identity), +and where any finite number collection {f(σp, µnσn)} is assumed to have Gaussian distribution with +mean 0 and covariance function κ((σp, µnσn), (σp, µnσn)′). The prediction µp for x = (σp, µn, σn) +given n observations (µi +p, σi +p, µi +n, σi +n) is then given by (see Rasmussen & Williams (2006)) +µp = +�κ(x1, x) +... +κ(xn, x)� +� +� +� +� +�� +(κ(x1, x1) +. . . +κ(x1, xn) +... +(κ(xn, x1) +. . . +κ(xn, xn) +� +�� +i,j ++ σ2 +εI +� +� +� +−1 � +�� +µ1 +p +... +µn +p +� +�� , +where we let xi = (σi +p, µi +n, σi +n). Here we take κ to be the squared exponential kernel, with parameters +estimated based on the standard loss function. The vector β and the intercept α are instead chosen +by minimization of the quantile loss function. +2In fact, several empirical studies provide evidence for this to be the case (see e.g. Friend & Blume (1975)). +3Observations are results of likelihood optimization, so 1% of outliers were excluded. + +8 +ACCEPTABLE BILATERAL GAMMA PARAMETERS +The linear estimates obtained for fm and fM are +fm(σp, µn, σn) = 0.0017 + 0.2029σp + 0.9951µn − 0.3711σn, +fM(σp, µn, σn) = 0.0017 + 0.2710σp + 1.0102µn − 0.2311σn. +Note, in particular, the negative relationship between σn and µp. +Similarly, for quantile GPR, ∂fm +∂σn are always negative, while ∂fM +∂σn are positive at all but two of 16 +representative points (table 1). +∂fM +∂σp +∂fM +∂µn +∂fM +∂σn +∂fm +∂σp +∂fm +∂µn +∂fm +∂σn +0.2667 +2.4704 +0.7577 +-0.0130 +2.0042 +-0.2421 +0.8691 +1.9402 +-1.3539 +1.1140 +1.8974 +-0.8361 +1.5243 +1.9553 +-1.1134 +1.4108 +1.9274 +-1.2346 +1.0459 +2.0254 +-0.4887 +0.5666 +1.9927 +-1.2635 +1.0867 +1.9956 +-1.0836 +0.8823 +2.0199 +-1.2220 +0.4639 +2.0065 +-1.4194 +0.6053 +2.0648 +-1.1568 +1.3013 +2.0509 +-1.4681 +1.2715 +2.0128 +-1.4149 +0.9669 +2.0019 +-0.2462 +0.4477 +1.9760 +-1.0806 +1.4434 +2.2522 +0.3978 +0.5052 +2.0026 +-0.5761 +0.9710 +1.9465 +-0.9840 +0.9472 +1.8900 +-0.8995 +1.0653 +1.9423 +-1.4702 +1.3075 +1.9230 +-0.9990 +0.9307 +1.9594 +-0.5941 +0.6044 +1.9087 +-1.0416 +1.3390 +2.0444 +-1.8664 +1.4529 +2.0287 +-1.5394 +0.8652 +1.9872 +-1.3001 +0.9281 +2.0499 +-1.0898 +1.1957 +2.0398 +-0.9586 +0.9027 +1.9967 +-1.3931 +0.9283 +1.9956 +-0.0906 +0.3913 +1.9830 +-0.9539 +Table 1. Boundaries gradients (estimated via Quantile GPR), at 16 representative points. +Alternatively, fm and fM can be obtained via distorted least square (Madan & Schoutens (2021)), +i.e. solving +min +f∈F +� +i +r2 +i +� +Ψ(qi) − Ψ +� +qi − 1 +n +�� +, +(3.2) +where, for every i, +ri := µp(i) − f(σp(i), µn(i), σn(i) +is the residual corresponding to the i-th observation, Ψ : [0, 1] → [0, 1] is a distribution function +(called a distortion) concave for fm and convex for fM, and qi is the i-th quantile of the residual’s +empirical distribution. +The idea behind 3.2 is as follows. First, Ψ defines a distorted expectation EΨ[X] of a random +variable X with distribution function F, as the Stjielties integral with respect to the distribution +function Ψ ◦ F: +EΨ[X] := +� +R +xdΨ(F(x)). +If Ψ is concave, lower values of X are weighted higher, thus implying EΨ[X] ≤ E[X], while the +opposite is true if Ψ is convex. Next, given observations {xi}n +i=0 of X, EΨ[X] is estimated by +n +� +i=1 +x(i) +� +Ψ(F(x(i))) − Ψ(F(x(i−1))) +� + +ACCEPTABLE BILATERAL GAMMA PARAMETERS +9 +where {xi}i=1,...,n is the ordered sample. If F is unknown, this estimator can be replaced by +n +� +i=1 +xi +� +Ψ (qi) − Ψ +� +qi − 1 +n +�� +, +so the loss function 3.2 corresponds to minimizing the estimated distorted expectation of the squared +residual. By the tower property of (nonlinear) conditional expectations, the solution to problem +3.2 minimizes the distorted squared distance to the estimate of EΨ[µp|σp, µn, σn] and so, for an +appropriate distortion, it can be thought as a lower/upper bound to the range of compensation µp +given the risks (σp, µn, σn). Following Madan & Schoutens (2021), we set γ = 0.75 and define, for +u ∈ [0, 1], +Ψ(u) = 1 − +� +1 − u +1 +1+γ +�1+γ +. +(3.3) +The linear estimates obtained for fm and fM obtained via distorted least square are +fm(σp, µn, σn) = 0.0024 + 0.1118σp + 0.9276µn − 0.2596σn, +fM(σp, µn, σn) = 0.0002 + 0.3604σp + 1.0196µn − 0.1798σn, +while the gradient at 16 representative points of the GPR estimates is shown in table 2. +∂fM +∂σp +∂fM +∂µn +∂fM +∂σn +∂fm +∂σp +∂fm +∂µn +∂fm +∂σn +-0.0887 +1.9864 +0.6572 +0.0790 +2.0073 +-0.1164 +1.4815 +1.9431 +-0.3117 +0.5231 +1.8825 +-1.8317 +1.1663 +1.9467 +-1.8564 +1.6447 +1.9112 +-0.6290 +0.8911 +2.0018 +-0.7190 +0.6395 +1.9916 +-1.1617 +1.2837 +1.9852 +-0.6655 +0.6932 +2.0128 +-1.5889 +0.7569 +1.9771 +-0.9006 +0.3673 +2.1029 +-1.5779 +1.6313 +2.0300 +-0.8589 +0.9376 +2.0127 +-2.0179 +0.7577 +1.9821 +-0.5676 +0.5676 +1.9736 +-0.8992 +0.5482 +2.0124 +-0.3588 +0.7854 +2.0058 +-0.0587 +1.2923 +1.9270 +-0.3932 +0.6114 +1.8902 +-1.5024 +1.5750 +1.9676 +-0.7003 +0.8115 +1.8921 +-1.7373 +0.9369 +1.9325 +-0.5317 +0.5394 +1.9128 +-1.1926 +1.8380 +2.0261 +-0.9785 +0.9812 +2.0308 +-2.3639 +1.2369 +1.9782 +-0.6258 +0.6211 +2.0585 +-1.6346 +1.2424 +2.0136 +-0.8368 +0.8070 +1.9979 +-1.5882 +0.6792 +1.9826 +-0.4806 +0.5497 +1.9767 +-0.7058 +Table 2. Boundaries gradients via Distorted GPR at 16 representative points. +Finally, we show in table 3 the percentages of observations represented by each of the 16 quantized +points. +µp +0.0694 +0.0208 +0.0343 +0.0167 +0.0685 +0.1428 +0.0308 +0.0119 +% +0.70 +0.76 +3.53 +10.49 +1.79 +0.72 +5.66 +14.99 +µp +0.0467 +0.0165 +0.0260 +0.0130 +0.0453 +0.1002 +0.0225 +0.0088 +% +2.11 +11.22 +5.52 +12.11 +2.87 +1.19 +8.33 +11.00 +Table 3. Percentage of observations represented by quantized point µp. + +10 +ACCEPTABLE BILATERAL GAMMA PARAMETERS +3.2. Implied Boundaries for Performance Measures. Table 4-15 show boundaries for µp, +Sharpe ratio and acceptability index at the 16 quantized points. 4 +As observed in Madan & Eberlein (2009), typically acceptability indexes based on the MIN- +MAXVAR distortion for returns on stocks and indexes are less than 0.15, with median values of +0.04 and more than 5% of observations at 0. These findings are consistent with the boundaries for +the acceptability index at the 16 representative points shown in table 13 for both short and long +positions. Note also that the acceptability index tends to be higher for long positions, which is also +to be expected. +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +0.0752 +0.0694 +0.0653 +0.0512 +0.0467 +0.0427 +0.0228 +0.0208 +0.0189 +0.0180 +0.0165 +0.0149 +0.0379 +0.0343 +0.0315 +0.0287 +0.0260 +0.0238 +0.0177 +0.0167 +0.0157 +0.0143 +0.0130 +0.0119 +0.0701 +0.0685 +0.0665 +0.0470 +0.0453 +0.0432 +0.1459 +0.1428 +0.1402 +0.1025 +0.1002 +0.0980 +0.0324 +0.0308 +0.0292 +0.0238 +0.0225 +0.0212 +0.0127 +0.0119 +0.0109 +0.0097 +0.0088 +0.0082 +Table 4. µp boundaries estimated via Quantile Regression, at 16 representative points. +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +0.0856 +0.0694 +0.0697 +0.0536 +0.0467 +0.0469 +0.0224 +0.0208 +0.0190 +0.0184 +0.0165 +0.0143 +0.0348 +0.0343 +0.0339 +0.0269 +0.0260 +0.0252 +0.0180 +0.0167 +0.0153 +0.0147 +0.0130 +0.0112 +0.0706 +0.0685 +0.0661 +0.0473 +0.0453 +0.0428 +0.1440 +0.1428 +0.1421 +0.1024 +0.1002 +0.0986 +0.0329 +0.0308 +0.0284 +0.0243 +0.0225 +0.0204 +0.0127 +0.0119 +0.0107 +0.0092 +0.0088 +0.0081 +Table 5. µp boundaries estimated via Quantile GPR, at 16 representative points. +3.3. Dimensional Analysis. As visible from tables 4-7, the different methodologies do not pro- +duce significantly different estimates for the boundaries fm and fM, and one may wonder if such +boundaries are indeed linear. As the boundaries are close to each others one way to assess if this is +the case is to compare the variance of the noise of a linear lower dimensional embedding with that +of a nonlinear one.5 Our results, summarized in table, provide evidence of the linearity of fm and +fM. +4The definition of Sharpe ratio adopted here is simply given by +µ +√ +t +σ , with µ = µp − µn, σ2 = σ2 +p + σ2 +n and +t = 250 business days. The acceptability index is defined in Madan & Eberlein (2009) as the maximal γ such that +the distorted expectation EΨγ [X] is nonnegative (or nonpositive for short position), where Ψγ is again taken as the +MINMAXVAR distortion. +5For the nonlinear embedding we utilize the Diffusion map algorithm, recently introduced in Coifman & Lafon +(2006). + +ACCEPTABLE BILATERAL GAMMA PARAMETERS +11 +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +0.0756 +0.0694 +0.0644 +0.0513 +0.0467 +0.0427 +0.0223 +0.0208 +0.0193 +0.0175 +0.0165 +0.0154 +0.0377 +0.0343 +0.0317 +0.0284 +0.0260 +0.0241 +0.0172 +0.0167 +0.0161 +0.0137 +0.0130 +0.0123 +0.0699 +0.0685 +0.0676 +0.0467 +0.0453 +0.0439 +0.1461 +0.1428 +0.1416 +0.1025 +0.1002 +0.0992 +0.0320 +0.0308 +0.0297 +0.0233 +0.0225 +0.0216 +0.0121 +0.0119 +0.0114 +0.0090 +0.0088 +0.0086 +Table 6. µp boundaries via Distorted LS at 16 representative points. +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +0.0751 +0.0694 +0.0694 +0.0511 +0.0467 +0.0453 +0.0245 +0.0208 +0.0167 +0.0182 +0.0165 +0.0143 +0.0409 +0.0343 +0.0274 +0.0325 +0.0260 +0.0193 +0.0172 +0.0167 +0.0161 +0.0138 +0.0130 +0.0120 +0.0694 +0.0685 +0.0664 +0.0475 +0.0453 +0.0420 +0.1446 +0.1428 +0.1410 +0.1021 +0.1002 +0.0982 +0.0324 +0.0308 +0.0284 +0.0233 +0.0225 +0.0212 +0.0122 +0.0119 +0.0114 +0.0091 +0.0088 +0.0086 +Table 7. µp boundaries via Distorted GPR at 16 representative points. +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +1.3983 +-0.1280 +-1.2170 +1.0860 +-0.2864 +-1.5225 +1.6557 +0.3176 +-0.9488 +1.9242 +0.6302 +-0.6800 +1.1930 +-0.2494 +-1.4179 +1.4190 +-0.0292 +-1.1870 +2.9743 +1.6717 +0.3023 +2.3051 +0.9589 +-0.2970 +2.4155 +0.9010 +-0.9568 +2.0173 +0.7184 +-0.9007 +2.5615 +0.5177 +-1.2321 +2.5085 +0.6926 +-1.0699 +2.1170 +0.7360 +-0.6640 +2.4731 +1.1418 +-0.2466 +3.1653 +1.8893 +0.5564 +3.5950 +2.2045 +1.0172 +Table 8. Sharpe ratio boundaries via Quantile Regression at 16 representative points. +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +4.0960 +-0.1253 +-0.0500 +1.7966 +-0.2803 +-0.2231 +1.4038 +0.3108 +-0.8727 +2.2708 +0.6168 +-1.1672 +-0.0615 +-0.2441 +-0.4023 +0.4234 +-0.0286 +-0.4702 +3.2179 +1.6361 +-0.1370 +2.7574 +0.9385 +-0.9590 +2.7685 +0.8818 +-1.3389 +2.1867 +0.7031 +-1.2120 +1.2525 +0.5067 +0.0483 +2.3867 +0.6779 +-0.5442 +2.4903 +0.7203 +-1.2574 +2.9818 +1.1174 +-0.9577 +3.0642 +1.8490 +0.2862 +2.7892 +2.1576 +0.9789 +Table 9. Sharpe ratio boundaries via Quantile GPR, at 16 representative points. + +12 +ACCEPTABLE BILATERAL GAMMA PARAMETERS +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +1.5043 +-0.1280 +-1.4702 +1.1167 +-0.2864 +-1.5371 +1.3725 +0.3176 +-0.6988 +1.4923 +0.6302 +-0.3014 +1.1352 +-0.2494 +-1.3296 +1.2542 +-0.0292 +-1.0275 +2.2277 +1.6717 +0.8526 +1.6831 +0.9589 +0.2273 +2.1671 +0.9010 +0.0706 +1.7605 +0.7184 +-0.3756 +2.7175 +0.5177 +-0.2952 +2.4707 +0.6926 +-0.1181 +1.7489 +0.7360 +-0.2058 +1.9404 +1.1418 +0.2373 +2.2607 +1.8893 +1.1881 +2.4546 +2.2045 +1.8048 +Table 10. Sharpe ratio boundaries via Distorted LS at 16 representative points. +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +1.3460 +-0.1253 +-0.1301 +1.0333 +-0.2803 +-0.7017 +2.7489 +0.3108 +-2.4183 +2.0463 +0.6168 +-1.2065 +2.3787 +-0.2441 +-3.0308 +3.3902 +-0.0286 +-3.5272 +2.1957 +1.6361 +0.8460 +1.7904 +0.9385 +-0.1558 +1.6968 +0.8818 +-1.0488 +2.3775 +0.7031 +-1.8278 +1.6345 +0.5067 +-0.6340 +2.1028 +0.6779 +-0.9154 +2.0997 +0.7203 +-1.2531 +1.9408 +1.1174 +-0.2121 +2.3064 +1.8490 +1.2318 +2.5459 +2.1576 +1.7381 +Table 11. Sharpe ratio boundaries via Distorted GPR at 16 representative points. +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +0.0570 +0.0036 +-0.0000 +0.0457 +-0.0000 +0.0000 +0.0667 +0.0188 +0.0000 +0.0750 +0.0283 +-0.0000 +0.0497 +-0.0000 +0.0000 +0.0578 +0.0065 +0.0000 +0.1132 +0.0637 +0.0123 +0.0877 +0.0394 +0.0000 +0.0949 +0.0378 +0.0000 +0.0809 +0.0306 +0.0000 +0.1027 +0.0249 +-0.0000 +0.1002 +0.0309 +0.0000 +0.0831 +0.0303 +0.0000 +0.0957 +0.0448 +-0.0000 +0.1207 +0.0740 +0.0210 +0.1367 +0.0858 +0.0376 +Table 12. Acceptability index boundaries via Quantile Regression at 16 representative +points. +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +0.1513 +-0.0039 +-0.0030 +0.0646 +-0.0097 +-0.0087 +0.0506 +0.0308 +-0.0109 +0.0824 +0.0413 +-0.0222 +-0.0150 +-0.0083 +-0.0017 +0.0174 +-0.0153 +-0.0008 +0.1171 +0.0588 +-0.0057 +0.1004 +0.0338 +-0.0338 +0.1010 +0.0487 +-0.0322 +0.0793 +0.0440 +-0.0255 +0.0454 +0.0188 +0.0006 +0.0870 +0.0248 +-0.0203 +0.0904 +0.0455 +-0.0261 +0.1089 +0.0403 +-0.0346 +0.1120 +0.0674 +0.0092 +0.1017 +0.0781 +0.0341 +Table 13. Acceptability index boundaries via Quantile GPR at 16 representative points. + +ACCEPTABLE BILATERAL GAMMA PARAMETERS +13 +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +0.0544 +0.0036 +-0.0000 +0.0408 +0.0000 +0.0000 +0.0504 +0.0189 +-0.0000 +0.0525 +0.0288 +-0.0000 +0.0413 +0.0000 +-0.0000 +0.0456 +0.0065 +0.0000 +0.0801 +0.0637 +0.0365 +0.0586 +0.0395 +0.0135 +0.0804 +0.0378 +0.0130 +0.0650 +0.0306 +-0.0000 +0.1017 +0.0249 +0.0013 +0.0920 +0.0308 +0.0070 +0.0641 +0.0305 +0.0024 +0.0702 +0.0451 +0.0166 +0.0813 +0.0734 +0.0493 +0.0877 +0.0856 +0.0695 +Table 14. Acceptability index boundaries via Distorted LS, at 16 representative points. +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +0.0490 +-0.0058 +-0.0036 +0.0369 +-0.0256 +-0.0099 +0.0995 +0.0866 +-0.0107 +0.0741 +0.0427 +-0.0222 +0.1140 +-0.0838 +-0.0102 +0.1328 +-0.1211 +-0.0027 +0.0796 +0.0588 +0.0290 +0.0650 +0.0337 +-0.0053 +0.0616 +0.0382 +-0.0322 +0.0861 +0.0662 +-0.0255 +0.0593 +0.0235 +-0.0188 +0.0765 +0.0335 +-0.0248 +0.0757 +0.0454 +-0.0262 +0.0701 +0.0403 +-0.0083 +0.0835 +0.0670 +0.0423 +0.0922 +0.0777 +0.0614 +Table 15. Acceptability index boundaries via Distorted GPR, at 16 representative points. +PCA +cumulative weight (in %) +Diffusion Map +cumulative weight (in %) +λ1 +2.7529 +68.82 +0.0113 +70.27 +λ2 +1.1778 +98.27 +0.0045 +98.58 +λ3 +0.0685 +99.98 +0.0002 +99.64 +λ4 +0.0009 +100.0 +0.0001 +100.0 +Table 16. Eigenvalues’s weights for PCA and diffusion map on the quantized dataset. +4. A Simple Modification of a Lucas Tree Economy +To formally link the risk-seeking behaviors observed above with those of prospects theory consider +the following modification of a Lucas tree economy (Lucas (1978)). There are two periods, and each +agent is endowed with a single risky asset with payoff Si at the end of period i, i = 0, 1. Assume +S1 = S0eG−L, where G and L are independent gamma distributed random variables. Suppose there +is a risk-free asset in zero net supply with risk-free rate rf, and that agents decide how mucht to +borrow/lend at time 0. Denoting such amount by ℓ, consumption Ci at period i, i = 0, 1, is +C0 = S0 + ℓ, C1 = S0eG−L − ℓerf . +Finally, setting X = G−L and s0 = log(S0), suppose that, for some 0 < ρ, β < 1, agents preferences +are described by +U(C0, C1) = (log(C0))1−ρ +1 − ρ ++ e−βE +�(log(C1))1−ρ +1 − ρ +11{s0+X≥0} − (− log(C1))1−ρ +1 − ρ +11{s0+X≤0} +� +. +(4.1) +This is a slight modification of the specification introduced in Kahneman & Tverski (1992) to pro- +vide a working framework that includes prospects theory experimental observations. In particular, +the investor is risk averse if and only if the log-return G − L is above the threshold s0, and this is +a behavior that cannot be captured by preferences over terminal wealth. + +14 +ACCEPTABLE BILATERAL GAMMA PARAMETERS +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +3 +4 +5 +6 +7 +8 +9 +10 +10-3 +(a) +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +-0.06 +-0.04 +-0.02 +0 +0.02 +0.04 +0.06 +0.08 +(b) +Figure 1. Equilibrium rate as a function of σp (a) and of µp (b). +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +(a) +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +-0.08 +-0.06 +-0.04 +-0.02 +0 +0.02 +0.04 +0.06 +0.08 +(b) +Figure 2. Equilibrium rate as a function of σn (a) and of µn (b). +In equilibrium, ℓ = 0, and so +s−ρ +0 +− erf e−β � +E[(s0 + X)−ρe−X11{s0+X≥0}] − E[(−s0 − X))−ρe−X11{s0+X≤0}] +� += 0, +and so the equilibrium interest rate re +f satisfies +re +f = β − ρ log(s0) − log +� +E[(s0 + X)−ρe−X11{s0+X≥0}] − E[(−s0 − X))−ρe−X11{s0+X≤0}] +� +. +(4.2) +For a risk averse individual, higher risks correspond to lower equilibrium risk free rate, as lending +becomes more attractive. Therefore, if the sign of ∂re +f/∂σn is negative, and that of ∂re +f/∂σp and +∂re +f/∂µn are positive, the simple setting here described provides an explanation of our empirical +findings. In general, it is possible to find values of (µp, σp, µn, σn) and of ρ such that this is indeed +the case. For instance, setting (µp, σp, µn, σn) = (0.03, 0.01, 0.03, 0.01), which are the average values +observed in the dataset above described, and setting ρ = 0.1 and β = 0.01, the value of re +f computed +via Montecarlo simulation as any of the variables (µp, σp, µn, σn) changes is shown in figures 1 and +2. As σp and/or σn increases the Montecarlo inegration estimate becomes less accurate as the +variance of X is higher, but the patterns in figures 1 and 2 confirm the behaviors above observed. + +ACCEPTABLE BILATERAL GAMMA PARAMETERS +15 +5. The Risks-Neutral Acceptance Set +In this section we analyze the “risk-neutral” acceptance set of quadruples (bp, cp, bn, cn) of BG +parameters estimated to option prices. The results reported are interesting on their own, but also +test the methodology employed to analyze the acceptance set of statistical parameters. +To better fit option prices, risk neutral log returns are modeled as ωt + Xt, where Xt is a BG +process, ω := r + log ((1 − bp)cp(1 + bn)cn) and r is the risk free rate. We calibrated the 10 sector +ETFs to the mid prices of options for four different maturities6, and obtained a risk neutral dataset +of 4812 observations. Figure 3 shows pairs (bp, cp) and (bn, cn) excluding 1% of outliers. Boundaries +for bp and bn in terms of (cp, bn, cn) and (cp, bn, cn) are estimated via quantile7 and distorted GPR. +8 Estimates are visualized in figure 4 and reported in table 17 and 18. +We observe that both quantile and distorted regression tend to break down in estimating the +boundaries of bp for large values of cp, mostly because this parameter ranges between 0 and 105, +with approximately 60% of the observations concentrated in the range [0, 30] and the remaining +ones being sparse (compare figure 3.A and 4.A) and corresponding to relatively small variations +in (bp, cn, bn). To avoid this issue, which - it is worth noting - does not occur when estimating +boundaries of bn (note that the range of observations for cn is [0, 50]), the regression algorithms for +the boundaries of bp are only based on observations corresponding to cp < 30. +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +0.0603 +0.0451 +0.0363 +0.0304 +0.0220 +0.0136 +0.0684 +0.0557 +0.0370 +0.0327 +0.0265 +0.0155 +0.0466 +0.0382 +0.0285 +0.0307 +0.0216 +0.0165 +0.0385 +0.0320 +0.0248 +0.0317 +0.0215 +0.0205 +0.0362 +0.0296 +0.0196 +0.0393 +0.0313 +0.0244 +0.0317 +0.0243 +0.0189 +0.0282 +0.0202 +0.0145 +0.0356 +0.0276 +0.0229 +0.0263 +0.0192 +0.0125 +0.0329 +0.0248 +0.0180 +0.0291 +0.0197 +0.0152 +Table 17. Boundaries for bp via quantile GPR at 16 representative points (with cp < 30). +Upper +Boundary +Observation +Lower +Boundary +Upper +Boundary +Observation +Lower +Boundary +0.0728 +0.0593 +0.0430 +0.2394 +0.1862 +0.1266 +0.0541 +0.0346 +0.0261 +0.2422 +0.1885 +0.1284 +0.2576 +0.1992 +0.1347 +0.2378 +0.1852 +0.1268 +0.2392 +0.1857 +0.1261 +0.2198 +0.1701 +0.1193 +0.2597 +0.2012 +0.1356 +0.2175 +0.1674 +0.1194 +0.2452 +0.1906 +0.1291 +0.2259 +0.1756 +0.1215 +0.2703 +0.2089 +0.1406 +0.2113 +0.1679 +0.1214 +0.2638 +0.2041 +0.1374 +0.2302 +0.1764 +0.1280 +Table 18. Boundaries for bn via quantile GPR at 16 representative points. +6Of all the traded maturities, the middle four were considered. Tickers considered are SPY, XLB, XLE, XLF, +XLI, XLK, XLP, XLU, XLV, XLY. Calibration was performed every 10 days between 1/01/2015 through 31/12/2020. +7For quantile GPR, the optimization was performed employing a quasi-Newton method, with the quantile loss +function approximated by S(x) = τx + α log(1 − e−x/α) as in Zheng (2011), with α = 10−4. +8In both cases, the hyperparameters of the kernel matrix K are estimated using the standard MSE loss function, +while α ∈ Rn and β ∈ R are computed so that β + Kα minimizes the quantile loss function. + +16 +ACCEPTABLE BILATERAL GAMMA PARAMETERS +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +105 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +(a) +0 +10 +20 +30 +40 +50 +60 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +(b) +Figure 3. Scatter plot of observed pairs (bp, cp) and (bn, cn) of risk neutral parameters. +0 +5 +10 +15 +20 +25 +30 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +(a) +0 +5 +10 +15 +20 +25 +30 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +(b) +Figure 4. Boundaries around randomly selected point (in red) via quantile GPR with +(τ = 0.05). +0 +5 +10 +15 +20 +25 +30 +35 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +(a) +0 +5 +10 +15 +20 +25 +30 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +(b) +Figure 5. Boundaries around randomly selected point (in red) via distorted GPR (γ = +0.75). + +ACCEPTABLE BILATERAL GAMMA PARAMETERS +17 +5.1. Speed Uncertainty. As mentioned, scale and shape parameters represent, respectively, limit +and market orders. Typically, professional traders place limit orders based on stable patterns and +strategies, and so the scale parameters arguably represent the phase of the economic cycle, and so +the speed parameters can be thought of as noisy responses to it.9 This would be, however, outside +of the theory of risk measures, as changing measure does not change speed parameters. Thus, +theoretical boundaries similar to those derived in the next section would require a notion of “speed +uncertainty”, similar to that of volatility uncertainty of G-Brownian motion (Peng (2006)). Such +notion can be implemented via nonlinear Levy processes (Neufeld & Nutz (2017)), according to +which, e.g., the ask price of a claim C = f(XT ), where X is a bilateral gamma process, is the +unique viscosity solutions of +� +ru(t, x) + supcp,cn∈Θ +�� +R\{0}[u(t, x + y) − u(t, x)]k(y)dy +� += ut(t, x), +u(0, x) = f(x) +where Θ ⊂ R2 is compact. There is however a large literature on the magnitude of the spread +between upper and lower valuations based on spectral risk measure, and since our empirical analysis +in the next sections is based on it, this approach was not further investigated. +5.2. An Equation for the Boundaries of Acceptable Risk Neutral Parameters. As men- +tioned in the introduction, the boundaries found for the risk neutral parameters are naturally linked +to acceptance sets implied by risk measures. In particular, given a fixed probability space (Ω, F, P) +and an asset’s bid and ask price processes {Bt}t≥0 and {At}t≥0, a version of the first fundamental +theorem of asset pricing with transaction costs asserts the existence of a probability measure Q and +a processes {St}t≥0 such that Q is equivalent to P, Bt ≤ St ≤ At for every t ≥ 0 and {e−rtSt}t≥0 is +a martingale under Q.10 Note that the measure Q is, approximately, a risk neutral measure in the +sense that the process St approximates the price at which one can buy and sell the asset. One can +then assume that the asset’s bid and ask prices be given by +B0 = inf +Q∈M EQ[S0e−r+ω+X1], A0 = sup +Q∈M +EQ[S0e−r+ω+X1]. +where M is a collection of probability measures that are equivalent to the statistical measure P. +Such collection is a financial primitive of the economy that, as anticipated in the introduction, +depends on regulator’s requirements for financial stability as well as trading, costs and incentives of +market operators, and a risk Z is deemed acceptable if EQ[Z] ≥ 0, ∀Q ∈ M. For our purposes, M +is defined as the set of measures associated to the spectral risk measure that arise from a distortion +Ψ ((one can employ e.g. the MINMAXVAR defined by 3.3). Bid and ask prices are then computed +as integrals of distorted probabilities of tail events (see Madan & Schoutens (2021)), and the higher +their distortion the higher size of the set M and the bid-ask spread. +Next, suppose that for given risk neutral parameters (ˆcp,ˆbn, ˆcn), the corresponding bp lies in the +interval [bp, bp]. It is natural to assume that +{Qbp}bp∈[bp,bp] ⊂ M. +(5.1) +where, for every bp ∈ [bp, bp], Qbp is a measure under which {Xt}t≥0 is a BG process with parameters +(bp, ˆcp,ˆbn, ˆcn). Note that, by proposition 6.1 in Kuchler & Tappe (2008), such a measure exists and +is equivalent to the risk neutral measure Q (and thus also to the statistical measure P). Since +(5.2) +inf +bp∈[bp,bp] +EQbp[e−r+ω+X1] = (1 − ˆbp)ˆcp +(1 − bp)ˆcp , +sup +bp∈[bp,bp] +EQbp[e−r+ω+X1] = (1 − ˆbp)ˆcp +(1 − bp)ˆcp , +9Diffusion map showed that more than 95% of the dataset variance is explained by two eigenvectors. +10For the existence of Q and the associated processes {St}t≥0 see Jouini & Kallal (1995) and Schachermeyer (2004). + +18 +ACCEPTABLE BILATERAL GAMMA PARAMETERS +where ω = r + log((1 − ˆbp)ˆcp(1 + ˆbn)ˆcn), 5.1 implies +(5.3) +B0 +S0 +≤ (1 − ˆbp)ˆcp +(1 − bp)ˆcp ≤ (1 − ˆbp)ˆcp +(1 − bp)ˆcp ≤ A0 +S0 +. +Further pushing 5.1 to be satisfied with an equality, we obtain the following relation for the upper +and lower boundaries bp and bp: +(5.4) +B0 +A0 += (1 − bp)ˆcp +(1 − bp)ˆcp +Note that, in 5.4, B0 and A0 are functions of ˆcp,ˆbn, ˆcn. In other words, the parameters ˆcp,ˆbn, ˆcn +are measures of economic activity and thus, together with the structural limits bp, bn, determine +bid and ask prices. Similarly, if ˆcp and ˆcn determine boundaries for bp and bn, +(5.5) +B0(ˆcp, ˆcn) +A0(ˆcp, ˆcn) = (1 − bp)ˆcp +(1 − bp)ˆcp +(1 + bn)ˆcn +(1 + bn)ˆcn . +5.3. Empirical Verifications. Typically, equations 5.4 and/or 5.5 are not satisfied, at least with +respect to daily closing bid-ask ratios. However, since large orders are executed over several days, +one can consider other distorted valuations, such as 5-days high/low prices. In general, one can +compare the size of M required for 5.1 to hold with typically observed acceptability indexes. To +do so, we let νbp denote the BG Levy measure with parameters (bp, ˆcp,ˆbn, ˆcn), and replace 5.1 with +{νbp}bp∈[bp,bp] ⊂ N, +(5.6) +The collection N is such that distorted rewards are defined by +µ = ω − +� ∞ +0 +G+(ν(ex − 1 < −a))da + +� ∞ +0 +(G−(ν(ex − 1 > a))da, +µ = ω − +� ∞ +0 +G−(ν(ex − 1 < −a))da + +� ∞ +0 +(G+(ν(ex − 1 > a))da +where ν is the Levy measure of X under Q and G+ and G− are (see Eberlein et al. (2013)) +G+(x) = x + 1 +c(1 − e−cx)1/(1+γ), G−(x) = x − 1 +c(1 − e−cx). +Then, as proved in Madan & Schoutens (2021), ˜ν ∈ N if and only if d˜ν +dν satisfies +(5.7) +S(λ) : = +� +R +�d˜ν +dν − λ +�+ +dν(x) ≤ Φ(λ), λ > 1, +˜S(λ) : = +� +R +� +λ − d˜ν +dν +�+ +dν(x) ≤ −˜Φ(λ), 0 ≤ λ < 1, +where Φ and ˜Φ are Fenchel conjugates of G+ and G− respectively, and are given by +Φ(λ) := 1 +c +� +−(1 − λ) log(u(λ)) + (1 − u(λ))1/(1+γ)� +, +−˜Φ(λ) := 1 +c[λ + (1 − λ) log(1 − λ)], +with u : (1, ∞) → (0, 1) defined as the unique solution of +u +(1 − u)γ/(1+γ) = (λ − 1)(1 + γ). + +ACCEPTABLE BILATERAL GAMMA PARAMETERS +19 +Next, we find requirements on c, γ for 5.6 to hold. Note that, for ˜ν = νbp, +S(λ) = +� ∞ +L(λ) +cp +x +� +e−x/bp − λe−x/ˆbp� +dx = cp +� +Ei +�L(λ) +bp +� +− λEi +� +L(λ) +ˆbp +�� +and, similarly, for ˜ν = νbp, +˜S(λ) = +� ∞ +˜L(λ) +cp +x +� +λe−x/ˆbp − e−x/bp +� +dx = cp +� +λEi +� +L(λ) +ˆbp +� +− Ei +�L(λ) +bp +�� +, +where Ei is the exponential integral function and +L(λ) = log(λ)bpˆbp +bp − ˆbp +, ˜L(λ) = −log(λ)ˆbpbp +ˆbp − bp +. +Lemma 5.1. Suppose bp > 0.55ˆbp. Then, νbp ∈ N holds for every bp ∈ [bp,ˆbp] if and only if +c ≤ lim +λ→1− +1 +˜S(λ) +(5.8) +Proof. Note that for every 0 < λ < 1 +−˜Φ′(λ) − ˜S′(λ) = − log(1 − λ) +c +− cp +� +Ei +� +L(λ) +ˆbp +� +− λe−L(λ)/ˆbp L′(λ) +L(λ) + e−L(λ)/bp L′(λ) +L(λ) +� += − log(1 − λ) +c +− cpEi +� +L(λ) +ˆbp +� +, +so that a stationary point ℓ of −˜Φ − ˜S must satisfy ccp = − log(1 − ℓ)/Ei +� +L(ℓ) +ˆbp +� +. Since +d +dλ +− log(1 − λ) +Ei +� +L(λ) +ˆbp +� += +Ei +� +L(ℓ) +ˆbp +� +1−λ +− e−L(λ)/ˆbp log(1−λ) +λ log(λ) +Ei +� +L(ℓ) +ˆbp +�2 +≤ e−L(λ)/ˆbp +log +� +1− +bp +(ˆbp−bp) log(λ) +� +1−λ +− log(1−λ) +λ log(λ) +Ei +� +L(ℓ) +ˆbp +�2 +, +the function −˜Φ − ˜S admits at most one stationary point in (0, 1) if +log +� +1 − +bp +(ˆbp−bp) log(λ) +� +1 − λ +− log(1 − λ) +λ log(λ) +< 0, ⇔ log(λ) +� +1 − (1 − λ) +1−λ +λ log(λ) +� +< +bp +(ˆbp − bp) +. +(5.9) +Since, for 0 < λ < 1, log(λ) +� +1 − (1 − λ) +1−λ +λ log(λ) +� +< 1.2, condition 5.9 holds if 0.55ˆbp < bp. Since +lim +λ→0+ −˜Φ(λ) = lim +λ→0+ ˜S(λ) = 0, +lim +λ→0+ +−˜Φ(λ) +˜S(λ) += ∞ +lim +λ→1− −˜Φ′(λ) = lim +λ→1− ˜S′(λ) = ∞, +lim +λ→1− +−˜Φ′(λ) +˜S′(λ) += 0, + +20 +ACCEPTABLE BILATERAL GAMMA PARAMETERS +it must be the case that if limλ→1− −˜Φ(λ) ≥ limλ→1− ˜S(λ), which is 5.8, then −˜Φ − ˜S admits +a positive maximum in (0, λ). Therefore, if 0.55ˆbp < bp and 5.8 are satisfied, −˜Φ − ˜S must be +nonnegative on (0, 1), since it would otherwise admit two stationary points. +□ +We note that 0.55ˆbp < bp for all the 16 representative points. The next two lemmas identify +necessary and sufficient conditions for the case bp ≥ ˆbp. +Lemma 5.2. Suppose bp ∈ [ˆbp, bp]. Then, it is necessary for νbp ∈ N to hold that +γ > bp − ˆbp +ˆbp +:= ˜γ +(5.10) +c ≤ 1 +cp +. +(5.11) +Proof. Note that +lim +λ→∞ Φ(λ) = lim +λ→∞ S(λ) = 0, +so, by l’Hopital’s theorem, Φ(λ) ≥ S(λ) implies +S′′(λ) +Φ′′(λ) = O(1) +as λ → ∞. Furthermore, using the implicit definition of u, +Φ′(λ) = 1 +c log(u(λ)), Φ′′(λ) = u′(λ) +cu(λ), S′(λ) = −cpEi +� +L(λ) +ˆbp +� +, S′′(λ) = cp +1 +λbp/(bp−ˆbp)+1 log(λ) +, +and, using implicit differentiation, +u′(λ) = +� +u2+1/γ +(1 − u + γ)(1 + γ)1/γ +� � +1 +(λ − 1) +�2+1/γ +∼ +� +1 +(γ)(1 + γ)1/γ +� � 1 +λ +�2+1/γ +. +We thus need +2 + 1 +γ < +bp +bp − ˆbp ++ 1 ⇒ γ > ˜γ. +For 5.11 simply note that +lim +λ→1+ +Φ(λ) +S(λ) = 1 +ccp +. +□ +Lemma 5.3. There is a function κp : (˜γ, ∞) → (0, ˜c], where +˜c := min +� +lim +λ→1− +1 +˜S(λ) +, 1 +cp +� +, +such that, for every γ that satisfies 5.10, νbp ∈ N if c < κp(γ) and νbp /∈ N if c > κp(γ). +Proof. Fix γ > ˜γ. As in the previous lemma, and since u does not depend on c, there is ℓ > 1 +independent of c such that for every λ > ℓ, +−(1 − λ) log(u(λ)) + (1 − u(λ))1/(1+γ) +Ei +� +L(λ) +bp +� +− λEi +� +L(λ) +ˆbp +� +≥ 1. +Hence, if c < ˜c, Φ(λ) > S(λ) for every λ > ℓ. Since Φ − S is continuous and decreasing in c for +every λ ∈ [1, ℓ], with limc→0 Φ − S = ∞, limc→∞ Φ − S = −S < 0, there is a bounded set of values + +ACCEPTABLE BILATERAL GAMMA PARAMETERS +21 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +(a) +Figure 6. The function Φ(λ)/S(λ) for the first of the 16 representative points, with c = ˜c +and assuming γ = ˜γ + 0.005 (blue) and γ = ˜γ (red). +c > 0 such that Φ(λ)−S(λ) ≥ 0 for every λ ∈ [1, ℓ]. Letting c denote the supremum of such values, +one can set κ(γ) = min{c, ˜c}. +□ +The function κp typically grows very fast, so that κp(γ) = ˜c for values of γ that are slightly larger +than ˜γ. For instance, figure 6 depicts the function Φ(λ/S(λ) for γ = ˜γ and γ′ = ˜γ + 0.005, and +for c = ˜c and the bilateral gamma parameters set as in the most representative of the quantized +points. In fact, this is the case for each of the 16 quantized points, as shown in table 19. +c +γ +˜γ +c +γ +˜γ +c +γ +˜γ +c +γ +˜γ +0.462 +0.357 +0.347 +0.212 +0.259 +0.219 +0.075 +0.400 +0.380 +0.138 +0.288 +0.258 +0.666 +0.263 +0.233 +0.130 +0.336 +0.296 +0.169 +0.256 +0.216 +0.061 +0.397 +0.377 +0.262 +0.252 +0.232 +0.126 +0.327 +0.287 +0.069 +0.484 +0.424 +0.039 +0.387 +0.357 +0.219 +0.209 +0.199 +0.096 +0.358 +0.328 +0.056 +0.529 +0.469 +0.040 +0.527 +0.467 +Table 19. Triples (˜c, γ, ˜γ) where γ is the minimal value ensuring 5.6 with c = ˜c. +Remark. Recalling that γ is similar to the acceptability index for probability distortions, while 10 +c +roughly corresponds to the maximum distorted frequencies (so higher c corresponds to smaller N), +we note that the values reported in 19 have the same magnitude and are consistent in general with +those typically seen in the literature (see for instance Eberlein et al. (2013), Elliot et al. (2022) and +Madan & Schoutens (2021)). +We observe, in particular, that +i. the three most representative points (top left corner of table 19), are consistent with the pair +(0.25, 0.25) used in Madan & Schoutens (2021) to estimate capital requirements (chapter +15.5.2), and that the relatively high values of γ is compensated by high values of c; +ii. for each triple, ˜c is higher, and in the less frequent cases close to, the value 0.01, which, as +shown in Elliot et al. (2022), generates higher returns compared to c = 1 and c = 0.25 for +a portfolio constructed by maximization of the lower valuation. +6. Conclusions +For an asset with (log) returns in the bilateral gamma class, a justification is provided, based on +expected utility theory, that risks from holding the asset can be decomposed into a three dimensional + +22 +ACCEPTABLE BILATERAL GAMMA PARAMETERS +vector of expected losses, variance of gains and variance of losses, while compensation for the risks +is given by expected gains. Evidence is then provided that moments of bilateral gamma returns lie +on a manifold with boundaries, and such boundaries are estimated via quantile and distorted linear +and nonlinear regressions. It is observed that they imply a positive relationship between expected +gains and variance of gains/expected losses, but a negative one between compensation and variance +of losses thus implying market’s operators being risk seekers in pure loss prospects. The claim that +such finding are compatible with the experimental evidence that constitute prospects theory is +then justified through a simple modification of Lucas Tree model. The analysis is corroborated +by performing a similar one to the case of risk neutral parameters, assuming a separate drift to +satisfy the martingale condition. An inverse relationship between shape and scale parameters of loss +and gain process is observed and a theoretical boundary for scale parameters, in line with certain +empirical observations, is described based on the theory of Conic finance. Finally, we observed +that our estimates of the boundaries are generally larger than those implied by regulatory capital +requirements. +7. Acknowledgment +This paper is a revised version of the second chapter of the author’s doctoral dissertation, which +was conducted under the supervision of Professor Dilip B. Madan at the Department of Mathematics +of the University of Maryland, College Park. +8. Appendix A: Assets Tickers +The list of tickers of the assets considered in the empirical analyses performed in this research +are reported in table 20 below. +a +aapl +abc +abt +adbe +adm +aep +afl +akam +all +amat +amp +amt +amzn +antm +aon +apa +apd +axp +ba +bac +bax +bby +bdx +ben +biib +bk +bmy +c +cah +cat +ccl +cf +chrw +cl +cma +cmcsa +cmi +cms +cof +cop +cost +crm +csco +ctsh +ctxs +cvs +cvx +d +de +dgx +dhr +dis +dov +duk +ebay +ecl +el +eog +eqt +etn +f +fcx +fdx +fitb +flr +fls +fslr +gd +ge +gild +gis +glw +gs +hal +hd +hes +hog +hon +hp +hpq +hum +ibm +ice +intc +isrg +itw +ivz +jci +jnj +jnpr +jpm +jwn +k +kim +klac +kmb +ko +kr +kss +lmt +lnc +low +m +ma +mcd +mck +mdt +met +mmc +mmm +mo +mrk +mro +ms +msft +mtb +mur +nem +nke +nov +nsc +ntap +nvda +nyt +orcl +oxy +payx +pcar +pfe +pg +ph +pnc +ppg +pru +pxd +rf +rhi +rl +rok +rrc +sbux +schw +slb +so +spg +spx +spy +stt +stz +syk +syy +t +tgt +tjx +tmo +trv +txn +txt +unh +unp +ups +usb +vix +vlo +vno +vz +wfc +whr +wmb +wmt +wy +x +xlb +xle +xlf +xli +xlk +xlp +xlu +xlv +xly +xom +xrx +Table 20 +References +Ali, M. 1975. 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Gradient descent algorithms for quantile regression with smooth approximation. +International Journal of Machine Learning and Cybernetics, 191–207. + diff --git a/EdE4T4oBgHgl3EQf6g7g/content/tmp_files/load_file.txt b/EdE4T4oBgHgl3EQf6g7g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..936b4c5d4d8ba33b17c13b6286c2db662465699d --- /dev/null +++ b/EdE4T4oBgHgl3EQf6g7g/content/tmp_files/load_file.txt @@ -0,0 +1,1988 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf,len=1987 +page_content='ACCEPTABLE BILATERAL GAMMA PARAMETERS YOSHIHIRO SHIRAI Department of Mathematics, University of Maryland, College Park Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The purpose of this paper is to utilize statistical methodologies to infer from market prices of assets and their derivatives the magnitude of the set of a measure M that defines acceptance sets of risky future cash flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' We assume that M contains the collection of bilateral gamma random variables, and estimate upper and lower boundaries of the compensation needed for a given bilateral gamma distributed future cash flow to be acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' We show that prospects theory provides a natural interpretation of the behaviors implied by such boundaries, which are not compatible with expected utility theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Boundaries for bilateral gamma risk neutral scale parameters for given speed parameters are also estimated and tested against market data and, in particular, comparisons are made with known empirical facts about the magnitude of the acceptance set of a common class of risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Introduction The definition of acceptable risks, based on the axiomatization of the concept of coherent risk measure given in Artzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (1999) and their convex generalization (Follmer & Schied (2002)), is a major recent advance in mathematical finance, as, among other applications, it provides an operative framework for superhedging in incomplete markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Starting from a monetary measure, such as Value at Risk, that only satisfies the basic requirements of monotonicity and cash invariance, practical considerations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' that the combined exposure of two trading desks ought to be less risky than that of the two desks taken separately, or that lack of liquidity may affect the future net worth of a single, large, position) lead one to require that a measure of risk also satisfy subadditivity and positive homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' A measure of risk ρ then defines a set Aρ of acceptable risks as those random variables X such that ρ(X) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Conversely, it is possible to show that given a cone A of acceptable risks, the functional ρ(X) = inf{m ∈ R : m + X ∈ A} (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1) satisfies monotonicity, cash invariance, subadditivity and positive homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Based on convex duality, a risk measure is also specified by a set of equivalent probability measures M as ρ(X) = inf Q∈M EQ[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2) The class M can be interpreted as the set of possible and credible macroeconomic/financial models, so that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2 is referred to as the robust representation of ρ, and risk measures become natural tools for the purpose of modeling uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' For convex risk measures, a penalty α(Q) is added to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2 to take into account that some models Q ∈ M may be more or less plausible than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' E-mail address: yshirai@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Date: January 16, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 60G18, 60G51, 91G20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Bilateral Gamma, Prospects Theory, Knightian Uncertainty, Risk Measures, Nonlinear Levy Processes, Diffusion Map, Quantile Regression, Distorted Regression, Gaussian Process Regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='05333v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='MF] 13 Jan 2023 2 ACCEPTABLE BILATERAL GAMMA PARAMETERS Typical examples of risk measures are those based on certainty equivalent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' such as the entropic risk measure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' which are known in general as utility-based shortfall risk measures and are defined by the acceptance set A = {X : E[u(X)] ≥ u(c)} for a given convex utility u and a threshold c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' and those obtained by modifying the tails of the underline statistical measure P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' such as the expected shortfall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' which are known in general as spectral risk measures and are defined by the Choquet integral ρ(X) = � ∞ 0 Ψ(P(X+ ≥ a))da − � ∞ 0 ˆΨ(P(X− ≥ a))da,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' where Ψ : [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 1] → [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 1] is increasing and convex and ˆΨ(u) = 1 − Ψ(1 − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' As the above examples confirm, relatively little is known in general about the set M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Note, however, that a risk measure is an expected value under a worst case scenario measure, and, as such, it defines a minimal current valuation (or maximal bid price) of the future cash flow X, while −ρ(−X) gives a maximal valuation (or the minimal ask price).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Assuming that market prices of traded assets are random variables whose distribution belong to a specific class and is determined by a set Θ ∈ RD of parameters, observed market prices imply specific boundaries for the set Θ and, in turn, for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' For instance, if M is (or contains) the class of normal random variables parameterized by pairs (µ, σ2) of mean and variance of assets returns, one can ask what are maximal and minimal bounds for µ given σ2 that are implied by historically observed pairs (µ, σ2) of traded assets, in turn estimated from market prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' These bounds are then naturally interpreted as structural limits for the reward µ given the risk σ2 that the economic system can offer without compromising its financial stability, as defined by the regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' To fix a reference framework, consider a market composed only of one risky asset with log return X and a riskless one in zero net supply with zero risk free rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, 1 = EQ[eX] = E[ηeX], (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3) where Q is a risk neutral measure, η the corresponding stochastic discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' If the distribution of X under the statistical measure P is parameterized by θ ∈ RD, and assuming the existence of a representative investor with utility U defined by a set of parameters ξ ∈ Rm, there is a function V : RD × Rm → R that evaluates to 1 at (θ, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Specifically (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Madan (2020a)), the risk neutral density (with respect to the log return) is given by h(x, θ, ξ) = U ′ ξ(ex)fθ(x) � R U ′ ξ(es)fθ(s)ds, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='4) where fθ is the statistical density of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Based on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='4, if the prospects offered by the risky asset suddenly deteriorate, 1with θ replaced by a riskier θ′, a decrease in the equilibrium risk free rate is needed to compensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In extreme cases, however, investors may no longer be allowed to hold such an asset which will be liquidated and may, ultimately, stop trading in some markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' As an example, one may think of pension funds, which are not allowed to hold speculative grade bonds, or to those asset classes, such as hedge funds, that are only reserved to institutional investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In the case of normal returns, as it is well known (Markovitz (1952), Tobin (1958), Sharpe (1964), Lintner (1965)), the efficient frontier essentially provides the upper limit for the reward µp given a risk defined by σ2, and also the lower one, as this is the upper limit for a short position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In general, however, this result lies on the assumption that investors have mean-variance preferences, and that, in particular, they are expected utility maximizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Empirical observations, on the other hand, have shown in many occasions that asset returns are not compatible with such axioms - a well known example being the equity premium puzzle, according to which U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' equity risk premia ACCEPTABLE BILATERAL GAMMA PARAMETERS 3 over Treasury Bills rates reflect an implausible level of aversion to risk under expected utility theory (Mehra & Prescott (1985)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' An alternative to expected utility theory, termed “prospects theory”, is based on a series of ex- periments conducted by psychologists D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Kahneman and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Tverski (Kahneman & Tverski (1979)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' One of their results, in particular, is that humans tend to be risk seekers rather than risk averse in the case of pure losses prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' For instance, the prospect of winning 1000 dollars with probability 1/2 and winning zero otherwise is generally dominated by the prospect of winning 500 dollars with probability 1, but the prospect of losing 1000 dollars with probability 1/2 and losing zero otherwise dominates the prospect of losing 500 dollars with probability 1, independently of initial wealth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Based on such evidence, one is then led to interpret an asset’s return as the sum of two prospects, one consisting of pure gains, and the other one of pure losses, and investors rank different assets’ returns based on the expectations and variances (µp, σ2 p, µn, σ2 n) of gains and losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In particular, higher variance of losses is compensated, ceteris paribus, by lower expectation µp of the gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The bilateral gamma distribution (Kuchler & Tappe (2008)) and its multivariate version (Madan (2020b)) provide a natural modeling framework for such a preference specification for several rea- sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Firstly, it is the difference of two independent gamma variates, interpretable as gains and losses, and it is completely specified by the vector (µp, σp, µn, σn) of their expected values and stan- dard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Secondly, even in a continuous time setting, the bilateral gamma process is the difference of two independent gamma processes, while, for instance, path realizations of diffusion processes have infinite variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Thirdly, the bilateral gamma distribution provides a very good fit to the (log) returns distribution implied by time series of returns and also by options prices (Kuchler & Tappe (2008)), which shows that it is more suitable than, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', the normal distribution for the purpose of modeling asset returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Finally, as shown below, the expected utility of an asset with bilateral gamma return X is a function F : (µp, σp, µn, σn) → E[u(X)], increasing in µp, and decreasing in σp, µn and σn, so that under expected utility theory variations in (σp, µn, σn) are compensated by variations of equal sign in µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Based on this considerations, we assume in this paper that the set of credible models M includes the set of bilateral gamma random variables, and we learn bounds fM, fm : (σp, µp, σn) → µp for µp given risks (σ2 p, µn, σ2 n) via quantile and/or distorted linear and/or Gaussian process regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' An interesting result obtained is that both boundaries are generally increasing in (σp, µp), but decreasing in σn, suggesting that investors, independently of their wealth, seek for lower (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' higher) risk when it comes to purely positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' negative) processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' We test the boundaries computed by assessing how well their implied performance measures (Sharpe ratio and acceptability index) compare with those typically observed in the financial markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Furthermore, we investigate the linearity of fM and fm by comparing the results of a linear lower dimensional embedding and a nonlinear one, and we show through a simple variation of a Lucas tree economy Lucas (1978) that the behaviors observed are indeed consistent with prospects theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Finally, we move our attention to the risk neutral world, based on the suggestive interpretation given in Madan (2020a) that, for bilateral gamma returns, the scale parameters (bp, bn) determine the structure of limit orders, while the speed parameters (cp, cn) determine that of market orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' It is then natural to assume that a relationship exists between the two pairs of parameters, in the sense that for given (cp, cn), the scale parameters (bp, bn) are bounded to a specific range, as the structure of market orders cannot be too independent from that of limit orders and viceversa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' As done for the statistical moments, the boundaries of such range are learned through quantile and distorted regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In this case, we determine theoretical boundaries as well based on the well known robust representation of spectral risk measures (Madan & Schoutens (2021)), and evidence is offered of their comparability with the empirically estimated ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' First we show that for bilateral gamma returns, risks and compensations are identified by the vector (σp, µn, σn) and µp respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Empirical 4 ACCEPTABLE BILATERAL GAMMA PARAMETERS observations are reported in section 3, and the variation on Lucas Tree model is presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Risk neutral parameters are analyzed in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Section 6 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Bilateral Gamma Returns 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' From Brownian Motion to Bilateral Gamma Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Given its central role in this paper, the construction and properties of the bilateral gamma process are reviewed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In Black & Scholes (1973), F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Black and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Scholes proposed to model the dynamics of log-returns as a Brownian motion (GBM), as prices exhibit exponential growth and on the assumption, rooted in an entropy maximization argument (Madan (2020a)), that log returns are asymptotically normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' However, returns exhibit heavier tails than those implied by the normal distribution (Fama (1965)) and frequent discontinuities in their path trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In addition, risk aversion results in periods of intense trading, determined by widespread selling in securities, alternating with lower activity ones, thus implying that returns’ quadratic variation is not linear in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' It also results in higher demand for out of the money (OTM) than for the corresponding OTM calls, generating a volatility smile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Another entropy maximization argument then suggests modeling economic time as a gamma process, and stock market log returns as Brownian motion evaluated at such gamma time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The resulting process, pioneered by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Madan and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Seneta (Madan & Seneta (1990)) and termed the variance gamma process, is a pure jump Levy process with infinite activity and finite variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In fact, such process is the difference of two i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' gamma processes, which naturally correspond to gains and losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Finally, motivated by the fact that downward jumps in prices are generally higher than upward ones, the bilateral gamma process is defined as the difference of two independent gamma processes with different shape and scale parameters (Kuchler & Tappe (2008)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The gains and losses increments have BG distribution βΓ(bp, cp, bn, cn), defined by the convolution βΓ(bp, cp, bn, cn) = Γ(bp, cp) ∗ Γ(−bn, cn), where bp, cp, bn, cn > 0 and, for α > 0, λ ∈ R, a Γ(λ, α)-distributed random variable has density f(x) = 1 Γ(α)|λ|α |x|α−1e−|x|/|λ| � 11{λ>0}(x)11{x>0}(x) + 11{λ<0}(x)11{x<0}(x) � , x ∈ R with Γ(α) the Gamma function at α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, expected value and standard deviation of gains and losses, denoted respectively by µp, σp, µn and σn, are given by µp = cpbp, σp = √cpbp, µn = cnbn, σn = √cnbn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' By the convolution theorem, the characteristic function of the increments in t units of time is ϕt(u) = (1 − iubp)−tcp (1 + iubn)−tcn , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1) and it follows easily from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 that BG densities are stable under convolution and are infinitely divisible, and so the BG process is a well defined Levy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' From formula 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 and the Levy- Khintchine representation we also deduce its Levy density to be k(x) = �cp x e−x/bp11{(0,∞)}(x) + cn |x|e−|x|/bn11{(−∞,0)}(x) � , x ∈ R which shows that a BG process enjoys the self decomposability property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 Then (see Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (207) and the references therein) a BG distributed random variable X is a limit law, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' there are centering and scaling constants {cn}n∈N and {bn}n∈N and a sequence {Zk}k∈N of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' random variables such that the distribution of bnSn + cn converges in distribution to X, where Sn = 1A random variable X is self decomposable if for any 0 < c < 1 there is an independent random variable XC such that X d= cX + Xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' A Levy process enjoys the self decomposability property if its increments are self decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' ACCEPTABLE BILATERAL GAMMA PARAMETERS 5 �n k=1 Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' This is a remarkable property, since if returns consist of some average of a large number of independent news or other type of influences, it is reasonable to expect that their distribution should be well approximated by a limit law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In the GBM case such law is the Gaussian, but, as noticed in Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (207), there is “no compelling economic motivation” for the scaling constants to be √n as in the classical central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Evidence of the goodness of fit of the BG density to returns distributions is presented in Kuchler & Tappe (2008), where, using data on DAX between 1996 and 1998, it is shown that the null hypothesis that the log returns distribution is in the BG class is not rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Furthermore, as proved in Kuchler & Tappe (2008), for all BG parameters there exists a measure Q equivalent to P such that, under Q, the discounted exponential BG process is a (local) martingale and an exponential BG process, and one typically succeed in fitting the option prices surface, at least for a single fixed maturity, through an exponential BG process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Bilateral Gamma Returns under Expected Utility Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The notion and character- izations of second order stochastic dominance (SSD) are recalled below (see Rothschild & Stiglitz (1970)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Given random variables X and Y , one says that X first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' second) order stochastically dominates Y , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' X ⪰1 Y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' X ⪰2 Y ) if and only if E[u(X)] ≥ E[u(Y )] for every increasing (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' increasing and concave) real valued function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Let X and Y be random variables with distribution functions F and G respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, X ⪰1 Y if and only if G(t) ≥ F(t) for every t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Let X and Y be random variables with distribution functions F and G respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, the following are equivalent (i) X ⪰2 Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (ii) there are random variables Z and ε such that Y ∼ X + Z + ε, Z ≤ 0 and E[ε|X + Z] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (iii) � t −∞ G(s)ds ≥ � s −∞ F(s)ds for every t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In addition, if E[X] = E[Y ], then the following are equivalent: (i) X ⪰2 Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (ii) there is a random variable ε such that Y ∼ X + ε and E[ε|X + Z] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (iii) E[u(X)] ≥ E[u(Y )] for every u concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Suppose X ⪰2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, E[X] ≥ E[Y ] and if E[X] = E[Y ] then V (X) ≤ V (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' That E[X] ≥ E[Y ] if X ⪰2 Y follows immediately from the fact that the identity is non decreasing and concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' If E[X] = E[Y ], then E[u(X)] ≥ E[u(Y )] for every u concave, and so, setting u(x) = −x2 + E[X], one obtains V (X) = E[X2 − E[X]] ≤ E[Y 2 − E[X]] = V [Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' □ Thus, for bilateral gamma returns, SSD implies higher expected gains and/or lower expected losses, and, for equal expected gains and losses, lower standard deviation of gains and/or losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' A partial converse of this statement is shown below, and is based on the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Let X and Y be random variables with densities f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' If the likelihood ratio f g is monotonically increasing, than X ⪰1 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' If the likelihood ratio is monotonically increasing on (−∞, x0) ∪ (x1, ∞) and decreasing on (x0, x1), with x0 < x1 ∈ R, then X ⪰2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' See Ali (1975) and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Let X and Y be two gamma distributed random variable with scale and shape parameters (b, c) and (b′, c′) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, (i) if b = b′, then c > c′ iff X ⪰2 Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (ii) if c = c′, then b > b′ iff X ⪰2 Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 6 ACCEPTABLE BILATERAL GAMMA PARAMETERS (ii) c c′ ≤ max(1, b′ b ) with strict inequality at least when b′ b = 1 iff X ⪰2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Based on showing that the assumptions of theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' See Ali (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Let X and Y be two gamma distributed random variables with scale and shape parameters (b, c) and (b′, c′) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, X second order stochastically dominates Y if E[X] ≥ E[Y ] and V [X] ≤ V [Y ] with at least one strict inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Similarly, −X second order stochastically dominates −Y if E[X] ≤ E[Y ] and V [X] ≤ V [Y ] with at least one strict inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Suppose E[X] ≥ E[Y ] and V [X] ≤ V [Y ] with at least one strict inequality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' bc ≥ b′c′ and b2c ≤ b′2c′ with at least one strict inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, c c′ ≥ b′ b , and b′ b = b′2c′ b2c bc b′c′ > 1 so X ⪰2 Y by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The result for −X and −Y follows from adapting Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='6 to the case of the negative of gamma distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Let X+, X−, Y +, Y − be four gamma distributed random variable with scale and shape parameters (bp, cp), (bn, cn), (b′ p, c′ p) and (b′ n, c′ n) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, X := X+ − X− second order stochastically dominates Y := Y +−Y − if E[X+] ≥ E[Y +], E[X−] ≤ E[Y −], V [X+] ≤ V [Y +], and V [X−] ≤ V [Y −] with exactly one strict inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Suppose for instance E[X+] > E[Y +].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='7, X+ ⪰2 Y +, and so, for all t ∈ [0, ∞) � t 0 F +(s) − G+(s)ds ≤ 0, where F + and G+ denote the cumulative distribution function of X+ and Y + respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, using Tonelli’s theorem, � t 0 F(s) − G(s)ds = � ∞ 0 � t 0 F +(s − ξ) − G+(s − ξ)dsdF −(ξ) ≤ 0, where F − is the (common) distribution of −X− and −Y −, and the conclusion follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The other cases are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' □ Based on the last corollary and transitivity of SSD, the observation that a positive variation in µp can compensate a positive variation in any among the upside volatility σp, the expected loss prospect µn or the downside volatility σn is evidence of investors’ risk seeking behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Note that µp is not, in general, a “reward” accessible to an investor holding the asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In fact, for a given time horizon T the expected return for holding the asset is the value µ(T) that satisfies S0eµ(T) = E[S0eXT ], and so the variation lim T↓0 µ(T) T = � R (ex − 1)k(x)dx = (1 − bp)−cp(1 + bn)−cn (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2) better serves this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Thus, we refer to µp as a “compensation” for the risks (σp, µn, σn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Log-Returns and Kelly’s Criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In the case log returns are assumed to be bilateral gamma variates, these results cannot hold anymore, since, for instance, an increase in σp and/or σn im- plies higher expected value of the return, and it cannot imply second order stochastic dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' However, a traditional assumption in the financial and economics literature, justified by some ev- idence (Arrow (1971)), is to assume that investors maximize log-returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In our context, such an assumption implies that an asset is preferred to another one if and only if the expected log-return is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' More generally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' for asset allocation problems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' logarithmic utility yields the best return in the long run,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' assuming the investor faces a long sequence of investment decisions (Kelly (1956),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' ACCEPTABLE BILATERAL GAMMA PARAMETERS 7 Merton (1969),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Cover (1991)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' but for an investor with a short/medium term horizon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' a logarithmic utility will not capture aversion to short term high volatility (Samuelson (1979)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' thus leading to consider a utility specification with a coefficient of relative risk aversion (CRRA) bounded below by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2 It then follows from the results of this section and proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 below that, for a reasonable utility specification such as u(log(·)), risks and their compensation are captured by (σp, µn, σn) and µp respectively even when log-returns belong to the bilateral gamma class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' A strictly increasing and concave function v ∈ C2 ((0, ∞)) has CRRA coefficient greater than 1 if and only if there is a strictly increasing and concave function u ∈ C2(R) such that v(x) = u(log(x)) for every x ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Suppose such a u exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, for all x ∈ (0, ∞), u′′(log(x)) ≥ 0 and u′(log(x)) < 0 xv′′(x) v′(x) = −x d2 dx2 u(log(x)) d dxu(log(x)) = 1 − u′′(log(x)) u′(log(x)) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' On the other hand, if v has CRRA bounded below by 1, then, setting u(y) = v(ey) for every y ∈ R, we obtain u′(y) = v′(ey)ey > 0 and u′′(y) = v′′(ey)e2y + v′(ey)ey ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The Acceptance Set 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Learning the Boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' As mentioned in the introduction, not all quadruples (µp, σp, µn, σn) can be traded, or, in other words, there are structural limits to how high and/or low is the level of rewards that can be offered for given risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In order to determine such limits, moments of gains and losses were estimated for 184 stocks (whose ticker is reported in appendix A) for the period 01/01/2008 to 31/12/2020 using one year of data for each estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3 Assuming the boundaries are defined by functions fm, fM : (σp, µn, σn) → µp, we find fM and fm by solving, respectively, min f∈F(1 − τM) � i [µp(i) − fM(σp(i), µn(i), σn(i))]+ − τM � i [µp(i) − fM(σp(i), µn(i), σn(i))]− , min f∈F(1 − τm) � i [µp(i) − fm(σp(i), µn(i), σn(i))]+ − τm � i [µp(i) − fm(σp(i), µn(i), σn(i))]− , where F is a suitable class of functions which is here assumed to be the class of linear Gaussian process (GPR) regressors, τM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='95 and τm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In our implementation of quantile GPR, the kernel hyperparameters were estimated using the standard loss function, while the regression coefficients are chosen to maximize the quantile loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Specifically, recall that GPR assumes µp = α + h(σp, µnσn)T β + f(σp, µnσn) + ε, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1) where ε is noise with variance σ2 ε, h is the map to features space (here assumed to be the identity), and where any finite number collection {f(σp, µnσn)} is assumed to have Gaussian distribution with mean 0 and covariance function κ((σp, µnσn), (σp, µnσn)′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The prediction µp for x = (σp, µn, σn) given n observations (µi p, σi p, µi n, σi n) is then given by (see Rasmussen & Williams (2006)) µp = �κ(x1, x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' κ(xn, x)� � � � � �� (κ(x1, x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' κ(x1, xn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (κ(xn, x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' κ(xn, xn) � �� i,j + σ2 εI � � � −1 � �� µ1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' µn p � �� , where we let xi = (σi p, µi n, σi n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Here we take κ to be the squared exponential kernel, with parameters estimated based on the standard loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The vector β and the intercept α are instead chosen by minimization of the quantile loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 2In fact, several empirical studies provide evidence for this to be the case (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Friend & Blume (1975)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 3Observations are results of likelihood optimization, so 1% of outliers were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 8 ACCEPTABLE BILATERAL GAMMA PARAMETERS The linear estimates obtained for fm and fM are fm(σp, µn, σn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0017 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2029σp + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9951µn − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3711σn, fM(σp, µn, σn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0017 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2710σp + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0102µn − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2311σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Note, in particular, the negative relationship between σn and µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Similarly, for quantile GPR, ∂fm ∂σn are always negative, while ∂fM ∂σn are positive at all but two of 16 representative points (table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' ∂fM ∂σp ∂fM ∂µn ∂fM ∂σn ∂fm ∂σp ∂fm ∂µn ∂fm ∂σn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2667 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='4704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='7577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0130 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='8691 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9402 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3913 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9830 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9539 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Boundaries gradients (estimated via Quantile GPR), at 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Alternatively, fm and fM can be obtained via distorted least square (Madan & Schoutens (2021)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' solving min f∈F � i r2 i � Ψ(qi) − Ψ � qi − 1 n �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2) where, for every i, ri := µp(i) − f(σp(i), µn(i), σn(i) is the residual corresponding to the i-th observation, Ψ : [0, 1] → [0, 1] is a distribution function (called a distortion) concave for fm and convex for fM, and qi is the i-th quantile of the residual’s empirical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The idea behind 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' First, Ψ defines a distorted expectation EΨ[X] of a random variable X with distribution function F, as the Stjielties integral with respect to the distribution function Ψ ◦ F: EΨ[X] := � R xdΨ(F(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' If Ψ is concave, lower values of X are weighted higher, thus implying EΨ[X] ≤ E[X], while the opposite is true if Ψ is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Next, given observations {xi}n i=0 of X, EΨ[X] is estimated by n � i=1 x(i) � Ψ(F(x(i))) − Ψ(F(x(i−1))) � ACCEPTABLE BILATERAL GAMMA PARAMETERS 9 where {xi}i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=',n is the ordered sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' If F is unknown, this estimator can be replaced by n � i=1 xi � Ψ (qi) − Ψ � qi − 1 n �� , so the loss function 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2 corresponds to minimizing the estimated distorted expectation of the squared residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' By the tower property of (nonlinear) conditional expectations, the solution to problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2 minimizes the distorted squared distance to the estimate of EΨ[µp|σp, µn, σn] and so, for an appropriate distortion, it can be thought as a lower/upper bound to the range of compensation µp given the risks (σp, µn, σn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Following Madan & Schoutens (2021), we set γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='75 and define, for u ∈ [0, 1], Ψ(u) = 1 − � 1 − u 1 1+γ �1+γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3) The linear estimates obtained for fm and fM obtained via distorted least square are fm(σp, µn, σn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0024 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1118σp + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9276µn − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2596σn, fM(σp, µn, σn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0002 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3604σp + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0196µn − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1798σn, while the gradient at 16 representative points of the GPR estimates is shown in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' ∂fM ∂σp ∂fM ∂µn ∂fM ∂σn ∂fm ∂σp ∂fm ∂µn ∂fm ∂σn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0887 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9864 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5497 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9767 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='7058 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Boundaries gradients via Distorted GPR at 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Finally, we show in table 3 the percentages of observations represented by each of the 16 quantized points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' µp 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0119 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='76 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='53 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='72 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='19 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='33 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='00 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Percentage of observations represented by quantized point µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 10 ACCEPTABLE BILATERAL GAMMA PARAMETERS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Implied Boundaries for Performance Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Table 4-15 show boundaries for µp, Sharpe ratio and acceptability index at the 16 quantized points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 4 As observed in Madan & Eberlein (2009), typically acceptability indexes based on the MIN- MAXVAR distortion for returns on stocks and indexes are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='15, with median values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='04 and more than 5% of observations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' These findings are consistent with the boundaries for the acceptability index at the 16 representative points shown in table 13 for both short and long positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Note also that the acceptability index tends to be higher for long positions, which is also to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Upper Boundary Observation Lower Boundary Upper Boundary Observation Lower Boundary 0.' metadata={'source': 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Quantile Regression, at 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Upper Boundary Observation Lower Boundary Upper Boundary Observation Lower Boundary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0697 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0536 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0467 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0081 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' µp boundaries estimated via Quantile GPR, at 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Dimensional Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' As visible from tables 4-7, the different methodologies do not pro- duce significantly different estimates for the boundaries fm and fM, and one may wonder if such boundaries are indeed linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' As the boundaries are close to each others one way to assess if this is the case is to compare the variance of the noise of a linear lower dimensional embedding with that of a nonlinear one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5 Our results, summarized in table, provide evidence of the linearity of fm and fM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 4The definition of Sharpe ratio adopted here is simply given by µ √ t σ , with µ = µp − µn, σ2 = σ2 p + σ2 n and t = 250 business days.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0086 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' µp boundaries via Distorted LS at 16 representative points.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='8490 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2862 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='7892 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9789 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Sharpe ratio boundaries via Quantile GPR, at 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 12 ACCEPTABLE BILATERAL GAMMA PARAMETERS Upper Boundary Observation Lower Boundary Upper Boundary Observation Lower Boundary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1280 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='4702 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2864 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='8048 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Sharpe ratio boundaries via Distorted LS at 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Upper Boundary Observation Lower Boundary Upper Boundary Observation Lower Boundary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1301 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5459 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1576 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='7381 Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Sharpe ratio boundaries via Distorted GPR at 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Upper Boundary Observation Lower Boundary Upper Boundary Observation Lower Boundary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0570 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0740 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0376 Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Acceptability index boundaries via Quantile Regression at 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Upper Boundary Observation Lower Boundary Upper Boundary Observation Lower Boundary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0097 0.' metadata={'source': 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+page_content='0493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0695 Table 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Acceptability index boundaries via Distorted LS, at 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Upper Boundary Observation Lower Boundary Upper Boundary Observation Lower Boundary 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0922 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0777 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0614 Table 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Acceptability index boundaries via Distorted GPR, at 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' PCA cumulative weight (in %) Diffusion Map cumulative weight (in %) λ1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='7529 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0113 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='27 λ2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1778 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0045 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='58 λ3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0685 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0002 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='64 λ4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0009 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0001 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0 Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Eigenvalues’s weights for PCA and diffusion map on the quantized dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' A Simple Modification of a Lucas Tree Economy To formally link the risk-seeking behaviors observed above with those of prospects theory consider the following modification of a Lucas tree economy (Lucas (1978)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' There are two periods, and each agent is endowed with a single risky asset with payoff Si at the end of period i, i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Assume S1 = S0eG−L, where G and L are independent gamma distributed random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Suppose there is a risk-free asset in zero net supply with risk-free rate rf, and that agents decide how mucht to borrow/lend at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Denoting such amount by ℓ, consumption Ci at period i, i = 0, 1, is C0 = S0 + ℓ, C1 = S0eG−L − ℓerf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Finally, setting X = G−L and s0 = log(S0), suppose that, for some 0 < ρ, β < 1, agents preferences are described by U(C0, C1) = (log(C0))1−ρ 1 − ρ + e−βE �(log(C1))1−ρ 1 − ρ 11{s0+X≥0} − (− log(C1))1−ρ 1 − ρ 11{s0+X≤0} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1) This is a slight modification of the specification introduced in Kahneman & Tverski (1992) to pro- vide a working framework that includes prospects theory experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In particular, the investor is risk averse if and only if the log-return G − L is above the threshold s0, and this is a behavior that cannot be captured by preferences over terminal wealth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 14 ACCEPTABLE BILATERAL GAMMA PARAMETERS 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='01 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='08 (b) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Equilibrium rate as a function of σp (a) and of µp (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='08 (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Equilibrium rate as a function of σn (a) and of µn (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In equilibrium, ℓ = 0, and so s−ρ 0 − erf e−β � E[(s0 + X)−ρe−X11{s0+X≥0}] − E[(−s0 − X))−ρe−X11{s0+X≤0}] � = 0, and so the equilibrium interest rate re f satisfies re f = β − ρ log(s0) − log � E[(s0 + X)−ρe−X11{s0+X≥0}] − E[(−s0 − X))−ρe−X11{s0+X≤0}] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2) For a risk averse individual, higher risks correspond to lower equilibrium risk free rate, as lending becomes more attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Therefore, if the sign of ∂re f/∂σn is negative, and that of ∂re f/∂σp and ∂re f/∂µn are positive, the simple setting here described provides an explanation of our empirical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In general, it is possible to find values of (µp, σp, µn, σn) and of ρ such that this is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' For instance, setting (µp, σp, µn, σn) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='01), which are the average values observed in the dataset above described, and setting ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='01, the value of re f computed via Montecarlo simulation as any of the variables (µp, σp, µn, σn) changes is shown in figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' As σp and/or σn increases the Montecarlo inegration estimate becomes less accurate as the variance of X is higher, but the patterns in figures 1 and 2 confirm the behaviors above observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' ACCEPTABLE BILATERAL GAMMA PARAMETERS 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The Risks-Neutral Acceptance Set In this section we analyze the “risk-neutral” acceptance set of quadruples (bp, cp, bn, cn) of BG parameters estimated to option prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The results reported are interesting on their own, but also test the methodology employed to analyze the acceptance set of statistical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' To better fit option prices, risk neutral log returns are modeled as ωt + Xt, where Xt is a BG process, ω := r + log ((1 − bp)cp(1 + bn)cn) and r is the risk free rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' We calibrated the 10 sector ETFs to the mid prices of options for four different maturities6, and obtained a risk neutral dataset of 4812 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Figure 3 shows pairs (bp, cp) and (bn, cn) excluding 1% of outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Boundaries for bp and bn in terms of (cp, bn, cn) and (cp, bn, cn) are estimated via quantile7 and distorted GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 8 Estimates are visualized in figure 4 and reported in table 17 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' We observe that both quantile and distorted regression tend to break down in estimating the boundaries of bp for large values of cp, mostly because this parameter ranges between 0 and 105, with approximately 60% of the observations concentrated in the range [0, 30] and the remaining ones being sparse (compare figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='A and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='A) and corresponding to relatively small variations in (bp, cn, bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' To avoid this issue, which - it is worth noting - does not occur when estimating boundaries of bn (note that the range of observations for cn is [0, 50]), the regression algorithms for the boundaries of bp are only based on observations corresponding to cp < 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Upper Boundary Observation Lower Boundary Upper Boundary Observation Lower Boundary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0603 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0291 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='0152 Table 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Boundaries for bp via quantile GPR at 16 representative points (with cp < 30).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1280 Table 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Boundaries for bn via quantile GPR at 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 6Of all the traded maturities, the middle four were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Tickers considered are SPY, XLB, XLE, XLF, XLI, XLK, XLP, XLU, XLV, XLY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Calibration was performed every 10 days between 1/01/2015 through 31/12/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 7For quantile GPR, the optimization was performed employing a quasi-Newton method, with the quantile loss function approximated by S(x) = τx + α log(1 − e−x/α) as in Zheng (2011), with α = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 8In both cases, the hyperparameters of the kernel matrix K are estimated using the standard MSE loss function, while α ∈ Rn and β ∈ R are computed so that β + Kα minimizes the quantile loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 16 ACCEPTABLE BILATERAL GAMMA PARAMETERS 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5 1 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5 (b) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Scatter plot of observed pairs (bp, cp) and (bn, cn) of risk neutral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 0 5 10 15 20 25 30 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='14 (a) 0 5 10 15 20 25 30 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5 (b) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Boundaries around randomly selected point (in red) via quantile GPR with (τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='14 (a) 0 5 10 15 20 25 30 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5 (b) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Boundaries around randomly selected point (in red) via distorted GPR (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' ACCEPTABLE BILATERAL GAMMA PARAMETERS 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Speed Uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' As mentioned, scale and shape parameters represent, respectively, limit and market orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Typically, professional traders place limit orders based on stable patterns and strategies, and so the scale parameters arguably represent the phase of the economic cycle, and so the speed parameters can be thought of as noisy responses to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9 This would be, however, outside of the theory of risk measures, as changing measure does not change speed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Thus, theoretical boundaries similar to those derived in the next section would require a notion of “speed uncertainty”, similar to that of volatility uncertainty of G-Brownian motion (Peng (2006)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Such notion can be implemented via nonlinear Levy processes (Neufeld & Nutz (2017)), according to which, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', the ask price of a claim C = f(XT ), where X is a bilateral gamma process, is the unique viscosity solutions of � ru(t, x) + supcp,cn∈Θ �� R\\{0}[u(t, x + y) − u(t, x)]k(y)dy � = ut(t, x), u(0, x) = f(x) where Θ ⊂ R2 is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' There is however a large literature on the magnitude of the spread between upper and lower valuations based on spectral risk measure, and since our empirical analysis in the next sections is based on it, this approach was not further investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' An Equation for the Boundaries of Acceptable Risk Neutral Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' As men- tioned in the introduction, the boundaries found for the risk neutral parameters are naturally linked to acceptance sets implied by risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In particular, given a fixed probability space (Ω, F, P) and an asset’s bid and ask price processes {Bt}t≥0 and {At}t≥0, a version of the first fundamental theorem of asset pricing with transaction costs asserts the existence of a probability measure Q and a processes {St}t≥0 such that Q is equivalent to P, Bt ≤ St ≤ At for every t ≥ 0 and {e−rtSt}t≥0 is a martingale under Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='10 Note that the measure Q is, approximately, a risk neutral measure in the sense that the process St approximates the price at which one can buy and sell the asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' One can then assume that the asset’s bid and ask prices be given by B0 = inf Q∈M EQ[S0e−r+ω+X1], A0 = sup Q∈M EQ[S0e−r+ω+X1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' where M is a collection of probability measures that are equivalent to the statistical measure P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Such collection is a financial primitive of the economy that, as anticipated in the introduction, depends on regulator’s requirements for financial stability as well as trading, costs and incentives of market operators, and a risk Z is deemed acceptable if EQ[Z] ≥ 0, ∀Q ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' For our purposes, M is defined as the set of measures associated to the spectral risk measure that arise from a distortion Ψ ((one can employ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' the MINMAXVAR defined by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Bid and ask prices are then computed as integrals of distorted probabilities of tail events (see Madan & Schoutens (2021)), and the higher their distortion the higher size of the set M and the bid-ask spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Next, suppose that for given risk neutral parameters (ˆcp,ˆbn, ˆcn), the corresponding bp lies in the interval [bp, bp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' It is natural to assume that {Qbp}bp∈[bp,bp] ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1) where, for every bp ∈ [bp, bp], Qbp is a measure under which {Xt}t≥0 is a BG process with parameters (bp, ˆcp,ˆbn, ˆcn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Note that, by proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 in Kuchler & Tappe (2008), such a measure exists and is equivalent to the risk neutral measure Q (and thus also to the statistical measure P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Since (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2) inf bp∈[bp,bp] EQbp[e−r+ω+X1] = (1 − ˆbp)ˆcp (1 − bp)ˆcp , sup bp∈[bp,bp] EQbp[e−r+ω+X1] = (1 − ˆbp)ˆcp (1 − bp)ˆcp , 9Diffusion map showed that more than 95% of the dataset variance is explained by two eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 10For the existence of Q and the associated processes {St}t≥0 see Jouini & Kallal (1995) and Schachermeyer (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 18 ACCEPTABLE BILATERAL GAMMA PARAMETERS where ω = r + log((1 − ˆbp)ˆcp(1 + ˆbn)ˆcn), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3) B0 S0 ≤ (1 − ˆbp)ˆcp (1 − bp)ˆcp ≤ (1 − ˆbp)ˆcp (1 − bp)ˆcp ≤ A0 S0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Further pushing 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 to be satisfied with an equality, we obtain the following relation for the upper and lower boundaries bp and bp: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='4) B0 A0 = (1 − bp)ˆcp (1 − bp)ˆcp Note that, in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='4, B0 and A0 are functions of ˆcp,ˆbn, ˆcn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In other words, the parameters ˆcp,ˆbn, ˆcn are measures of economic activity and thus, together with the structural limits bp, bn, determine bid and ask prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Similarly, if ˆcp and ˆcn determine boundaries for bp and bn, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5) B0(ˆcp, ˆcn) A0(ˆcp, ˆcn) = (1 − bp)ˆcp (1 − bp)ˆcp (1 + bn)ˆcn (1 + bn)ˆcn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Empirical Verifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Typically, equations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='4 and/or 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5 are not satisfied, at least with respect to daily closing bid-ask ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' However, since large orders are executed over several days, one can consider other distorted valuations, such as 5-days high/low prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In general, one can compare the size of M required for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 to hold with typically observed acceptability indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' To do so, we let νbp denote the BG Levy measure with parameters (bp, ˆcp,ˆbn, ˆcn), and replace 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1 with {νbp}bp∈[bp,bp] ⊂ N, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='6) The collection N is such that distorted rewards are defined by µ = ω − � ∞ 0 G+(ν(ex − 1 < −a))da + � ∞ 0 (G−(ν(ex − 1 > a))da, µ = ω − � ∞ 0 G−(ν(ex − 1 < −a))da + � ∞ 0 (G+(ν(ex − 1 > a))da where ν is the Levy measure of X under Q and G+ and G− are (see Eberlein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (2013)) G+(x) = x + 1 c(1 − e−cx)1/(1+γ), G−(x) = x − 1 c(1 − e−cx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, as proved in Madan & Schoutens (2021), ˜ν ∈ N if and only if d˜ν dν satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='7) S(λ) : = � R �d˜ν dν − λ �+ dν(x) ≤ Φ(λ), λ > 1, ˜S(λ) : = � R � λ − d˜ν dν �+ dν(x) ≤ −˜Φ(λ), 0 ≤ λ < 1, where Φ and ˜Φ are Fenchel conjugates of G+ and G− respectively, and are given by Φ(λ) := 1 c � −(1 − λ) log(u(λ)) + (1 − u(λ))1/(1+γ)� , −˜Φ(λ) := 1 c[λ + (1 − λ) log(1 − λ)], with u : (1, ∞) → (0, 1) defined as the unique solution of u (1 − u)γ/(1+γ) = (λ − 1)(1 + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' ACCEPTABLE BILATERAL GAMMA PARAMETERS 19 Next, we find requirements on c, γ for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='6 to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Note that, for ˜ν = νbp, S(λ) = � ∞ L(λ) cp x � e−x/bp − λe−x/ˆbp� dx = cp � Ei �L(λ) bp � − λEi � L(λ) ˆbp �� and, similarly, for ˜ν = νbp, ˜S(λ) = � ∞ ˜L(λ) cp x � λe−x/ˆbp − e−x/bp � dx = cp � λEi � L(λ) ˆbp � − Ei �L(λ) bp �� , where Ei is the exponential integral function and L(λ) = log(λ)bpˆbp bp − ˆbp , ˜L(λ) = −log(λ)ˆbpbp ˆbp − bp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Suppose bp > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='55ˆbp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, νbp ∈ N holds for every bp ∈ [bp,ˆbp] if and only if c ≤ lim λ→1− 1 ˜S(λ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Note that for every 0 < λ < 1 −˜Φ′(λ) − ˜S′(λ) = − log(1 − λ) c − cp � Ei � L(λ) ˆbp � − λe−L(λ)/ˆbp L′(λ) L(λ) + e−L(λ)/bp L′(λ) L(λ) � = − log(1 − λ) c − cpEi � L(λ) ˆbp � , so that a stationary point ℓ of −˜Φ − ˜S must satisfy ccp = − log(1 − ℓ)/Ei � L(ℓ) ˆbp � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Since d dλ − log(1 − λ) Ei � L(λ) ˆbp � = Ei � L(ℓ) ˆbp � 1−λ − e−L(λ)/ˆbp log(1−λ) λ log(λ) Ei � L(ℓ) ˆbp �2 ≤ e−L(λ)/ˆbp log � 1− bp (ˆbp−bp) log(λ) � 1−λ − log(1−λ) λ log(λ) Ei � L(ℓ) ˆbp �2 , the function −˜Φ − ˜S admits at most one stationary point in (0, 1) if log � 1 − bp (ˆbp−bp) log(λ) � 1 − λ − log(1 − λ) λ log(λ) < 0, ⇔ log(λ) � 1 − (1 − λ) 1−λ λ log(λ) � < bp (ˆbp − bp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9) Since, for 0 < λ < 1, log(λ) � 1 − (1 − λ) 1−λ λ log(λ) � < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2, condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='9 holds if 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='55ˆbp < bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Since lim λ→0+ −˜Φ(λ) = lim λ→0+ ˜S(λ) = 0, lim λ→0+ −˜Φ(λ) ˜S(λ) = ∞ lim λ→1− −˜Φ′(λ) = lim λ→1− ˜S′(λ) = ∞, lim λ→1− −˜Φ′(λ) ˜S′(λ) = 0, 20 ACCEPTABLE BILATERAL GAMMA PARAMETERS it must be the case that if limλ→1− −˜Φ(λ) ≥ limλ→1− ˜S(λ), which is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='8, then −˜Φ − ˜S admits a positive maximum in (0, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Therefore, if 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='55ˆbp < bp and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='8 are satisfied, −˜Φ − ˜S must be nonnegative on (0, 1), since it would otherwise admit two stationary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' □ We note that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='55ˆbp < bp for all the 16 representative points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The next two lemmas identify necessary and sufficient conditions for the case bp ≥ ˆbp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Suppose bp ∈ [ˆbp, bp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Then, it is necessary for νbp ∈ N to hold that γ > bp − ˆbp ˆbp := ˜γ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='10) c ≤ 1 cp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Note that lim λ→∞ Φ(λ) = lim λ→∞ S(λ) = 0, so, by l’Hopital’s theorem, Φ(λ) ≥ S(λ) implies S′′(λ) Φ′′(λ) = O(1) as λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Furthermore, using the implicit definition of u, Φ′(λ) = 1 c log(u(λ)), Φ′′(λ) = u′(λ) cu(λ), S′(λ) = −cpEi � L(λ) ˆbp � , S′′(λ) = cp 1 λbp/(bp−ˆbp)+1 log(λ) , and, using implicit differentiation, u′(λ) = � u2+1/γ (1 − u + γ)(1 + γ)1/γ � � 1 (λ − 1) �2+1/γ ∼ � 1 (γ)(1 + γ)1/γ � � 1 λ �2+1/γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' We thus need 2 + 1 γ < bp bp − ˆbp + 1 ⇒ γ > ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' For 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='11 simply note that lim λ→1+ Φ(λ) S(λ) = 1 ccp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' There is a function κp : (˜γ, ∞) → (0, ˜c], where ˜c := min � lim λ→1− 1 ˜S(λ) , 1 cp � , such that, for every γ that satisfies 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='10, νbp ∈ N if c < κp(γ) and νbp /∈ N if c > κp(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Fix γ > ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' As in the previous lemma, and since u does not depend on c, there is ℓ > 1 independent of c such that for every λ > ℓ, −(1 − λ) log(u(λ)) + (1 − u(λ))1/(1+γ) Ei � L(λ) bp � − λEi � L(λ) ˆbp � ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Hence, if c < ˜c, Φ(λ) > S(λ) for every λ > ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Since Φ − S is continuous and decreasing in c for every λ ∈ [1, ℓ], with limc→0 Φ − S = ∞, limc→∞ Φ − S = −S < 0, there is a bounded set of values ACCEPTABLE BILATERAL GAMMA PARAMETERS 21 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 (a) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The function Φ(λ)/S(λ) for the first of the 16 representative points, with c = ˜c and assuming γ = ˜γ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='005 (blue) and γ = ˜γ (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' c > 0 such that Φ(λ)−S(λ) ≥ 0 for every λ ∈ [1, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Letting c denote the supremum of such values, one can set κ(γ) = min{c, ˜c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' □ The function κp typically grows very fast, so that κp(γ) = ˜c for values of γ that are slightly larger than ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' For instance, figure 6 depicts the function Φ(λ/S(λ) for γ = ˜γ and γ′ = ˜γ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='005, and for c = ˜c and the bilateral gamma parameters set as in the most representative of the quantized points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' In fact, this is the case for each of the 16 quantized points, as shown in table 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' c γ ˜γ c γ ˜γ c γ ˜γ c γ ˜γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='357 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='527 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='467 Table 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Triples (˜c, γ, ˜γ) where γ is the minimal value ensuring 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='6 with c = ˜c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Recalling that γ is similar to the acceptability index for probability distortions, while 10 c roughly corresponds to the maximum distorted frequencies (so higher c corresponds to smaller N), we note that the values reported in 19 have the same magnitude and are consistent in general with those typically seen in the literature (see for instance Eberlein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (2013), Elliot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (2022) and Madan & Schoutens (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' We observe, in particular, that i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' the three most representative points (top left corner of table 19), are consistent with the pair (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='25) used in Madan & Schoutens (2021) to estimate capital requirements (chapter 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='2), and that the relatively high values of γ is compensated by high values of c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' for each triple, ˜c is higher, and in the less frequent cases close to, the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='01, which, as shown in Elliot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' (2022), generates higher returns compared to c = 1 and c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='25 for a portfolio constructed by maximization of the lower valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Conclusions For an asset with (log) returns in the bilateral gamma class, a justification is provided, based on expected utility theory, that risks from holding the asset can be decomposed into a three dimensional 22 ACCEPTABLE BILATERAL GAMMA PARAMETERS vector of expected losses, variance of gains and variance of losses, while compensation for the risks is given by expected gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Evidence is then provided that moments of bilateral gamma returns lie on a manifold with boundaries, and such boundaries are estimated via quantile and distorted linear and nonlinear regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' It is observed that they imply a positive relationship between expected gains and variance of gains/expected losses, but a negative one between compensation and variance of losses thus implying market’s operators being risk seekers in pure loss prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The claim that such finding are compatible with the experimental evidence that constitute prospects theory is then justified through a simple modification of Lucas Tree model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The analysis is corroborated by performing a similar one to the case of risk neutral parameters, assuming a separate drift to satisfy the martingale condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' An inverse relationship between shape and scale parameters of loss and gain process is observed and a theoretical boundary for scale parameters, in line with certain empirical observations, is described based on the theory of Conic finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Finally, we observed that our estimates of the boundaries are generally larger than those implied by regulatory capital requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Acknowledgment This paper is a revised version of the second chapter of the author’s doctoral dissertation, which was conducted under the supervision of Professor Dilip B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Madan at the Department of Mathematics of the University of Maryland, College Park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Appendix A: Assets Tickers The list of tickers of the assets considered in the empirical analyses performed in this research are reported in table 20 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='xrx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='Table 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='References ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content='Ali,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Stochastic Dominance and Portfolio Analysis.' 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Carr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', Geman, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', Madan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', & Yor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Self-Decomposabilitt and option pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Mathe- matical Finance, 17, 31–57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Coifman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', & Lafon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Diffusion maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Harmon.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Eberlein, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', Madan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', Pistorius, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', & Yor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' A Simple Stochastic Rate Model for Rate Equity Hybrid Products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Applied Mathematical Finance, 20(5), 461–488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Elliot, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', Madan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', & Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' High Dimensional Markov Trading of a Single Stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' SSRN Electronic Journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Fama, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The Behavior of Stock Market Prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Journal of Business, 38, 34–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Follmer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', & Schied, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Convex MEasures of Risk and Trading Constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Finance and Stochastics, 6(4), 429–447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Friend, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=', & Blume, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' The Review of Economic Studies, 25, 65–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Zheng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' Gradient descent algorithms for quantile regression with smooth approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} +page_content=' International Journal of Machine Learning and Cybernetics, 191–207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQf6g7g/content/2301.05333v1.pdf'} diff 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Samad and Sakib Abrar +Department of Computer Science +Tennessee State University +Nashville, TN, USA +msamad@tnstate.edu +January 3, 2023 +ABSTRACT +The latent space of autoencoders has been improved for clustering image data by jointly learning a +t-distributed embedding with a clustering algorithm inspired by the neighborhood embedding con- +cept proposed for data visualization. However, multivariate tabular data pose different challenges +in representation learning than image data, where traditional machine learning is often superior to +deep tabular data learning. In this paper, we address the challenge of learning tabular data in con- +trast to image data and present a novel Gaussian Cluster Embedding in Autoencoder Latent Space +(G-CEALS) algorithm by replacing t-distributions with multivariate Gaussian clusters. Unlike cur- +rent methods, the proposed method defines the Gaussian embedding and the target cluster distribu- +tion independently to accommodate any clustering algorithm in representation learning. A trained +G-CEALS model extracts a quality embedding for unseen test data. Based on the embedding clus- +tering accuracy, the average rank of the proposed G-CEALS method is 1.4 (0.7), which is superior +to all eight baseline clustering and cluster embedding methods on seven tabular data sets. This pa- +per shows one of the first algorithms to jointly learn embedding and clustering for improving the +representation of multivariate tabular data in downstream clustering. +Keywords embedding clustering, tabular data, Gaussian clusters, autoencoder, representation learning, multivariate +distribution +1 +Introduction +Deep learning has replaced traditional machine learning in many data-intensive research and applications due to its +ability to perform concurrent and efficient representation learning and classification. This concurrent learning approach +outperforms traditional machine learning that requires handcrafted features to perform supervised classification [1, 2]. +However, representation learning via supervisory signals from ground truth labels may be prone to overfitting [3] and +adversarial attacks [4]. Moreover, human annotations for supervised representation learning and classification may +not be available in all data domains or for all data samples. To address these pitfalls, representation learning via +unsupervised clustering algorithms may be a strong alternative to supervised learning methods. +The limitation of supervised representation learning may be overcome using self-supervision or pseudo labels that do +not require human-annotated supervisory signals [5, 6]. In a self-supervised autoencoder, the objective is to preserve +all information of input data in a low-dimensional embedding for data reconstruction. However, embeddings for data +reconstruction do not emphasize representations essential for downstream classification or clustering tasks. Therefore, +unsupervised methods have been proposed for jointly learning embedding with clustering to yield clustering friendly +representations [7, 8, 9, 10, 11]. The existing cluster embedding literature suggests several strict assumptions about +clustering algorithms (k-means), cluster distributions (t-distribution), and data modality (image data). While deep +representation learning of image data is well studied using convolutional neural networks (CNN), deep learning has +not seen much success with structured tabular data. There is strong evidence in the literature that traditional machine +arXiv:2301.00802v1 [cs.LG] 2 Jan 2023 + +A PREPRINT - JANUARY 3, 2023 +learning still outperforms deep models in learning tabular data [12, 13, 14, 15, 16]. In this paper, we review the +assumptions made in the cluster embedding literature and revise those assumptions for the representation learning of +tabular data. Accordingly, a novel joint learning framework is proposed considering the architectural and algorithmic +differences in learning image and tabular data. +The remainder of this manuscript is organized as follows. Section 2 provides a review of the state-of-the-art literature +on deep cluster embedding. Section 3 introduces tabular data with some theoretical underpinnings of neighborhood +embedding and cluster embedding in support of our proposed representation learning framework. Section 4 outlines +the proposed joint cluster embedding framework to obtain a quality representation of tabular data for downstream +clustering or classification. Section 5 summarizes the tabular data sets and experiments for evaluating the proposed +joint learning framework. Section 6 provides the results following the experiments and compares our proposed method +with similar methods in the literature. Section 7 summarizes the findings with additional insights into the results and +limitations. The paper concludes in Section 8. +2 +Related work +One of the earliest studies on cluster embedding, Deep Embedded Clustering (DEC) [7], is inspired by the seminal +work on t-distributed stochastic neighborhood embedding (t-SNE) [17]. The DEC approach first trains a deep autoen- +coder by minimizing the data reconstruction loss. The trained encoder part (excluding the decoder) is then fine-tuned +by minimizing the Kullback-Leibler (KL) divergence between a t-distributed cluster distribution (Q) on the embed- +ding and a target distribution (P). The target distribution is obtained via a closed-form solution by taking the first +derivative of the KL divergence loss between P and Q distributions with respect to P and equating it to zero. There- +fore, the assumption of t-distribution holds for both Q and P distributions in similar work. The k-means clustering in +the DEC approach is later replaced by spectral clustering to improve the quality of embedding in terms of clustering +performance [18]. The DEC approach is also enhanced by an improved DEC (IDEC) framework [8]. In IDEC, the +autoencoder reconstruction loss and the KL divergence loss are jointly minimized to update the weights of a deep +autoencoder and produce the embedding. Similar approaches, including t-distributions, k-means clustering, and KL +divergence loss, are adopted in joint embedding and cluster learning (JECL) for multimodal representation learning +of text-image data pairs [19]. The Deep Clustering via Joint Convolutional Autoencoder (DEPICT) approach learns +image embedding via a de-noising autoencoder [20]. The embedding is mapped to a softmax function to obtain a clus- +ter distribution or likelihood (Q) instead of assuming a distribution. Following a series of mathematical derivations +and assumptions, their final learning objective includes a cross-entropy loss involving P and Q distributions and an +embedding reconstruction loss for each layer of the convolutional autoencoder. +A general trend in the cluster embedding literature shows that K-means is the most common clustering method [7, 8, +10, 9, 21, 19, 20]. The assumption of t-distributed cluster embedding made in the DEC method [7] continues to appear +in the literature [22, 23, 8, 18, 19, 24] without any alternatives. The assumption of t-distribution is originally made in +the t-SNE algorithm for data visualization using neighborhood embedding maps [17]. We argue that the assumptions of +neighborhood embedding for data visualization are not aligned with the requirements of cluster embedding. Moreover, +cluster embedding methods proposed in the literature are invariably evaluated on benchmark image data sets. The +methods for image learning may not be optimal or even ready to learn tabular data representations. To the best of our +knowledge, similar cluster embedding methods have not been studied on multivariate tabular data. +2.1 +Contributions +This paper is one of the first to investigate the performance of joint cluster embedding methods on tabular data. The +limitations of state-of-the-art joint cluster embedding methods are addressed to contribute a new cluster embedding +algorithm as follows. First, we replace the current assumption of t-distributed embedding with a mixture of mul- +tivariate Gaussian distributions for multivariate tabular data by providing a theoretical underpinning for this choice. +Second, a new cluster embedding algorithm is proposed using multivariate Gaussian distributions that can jointly learn +distributions with any clustering algorithm. Third, we define the target cluster distribution on the tabular data space +instead of deriving it from the embedding because traditional machine learning of tabular data is still superior to deep +learning and can add complementary benefits to the embedding learned via an autoencoder. Therefore, our embedding +and target distributions are independent of each other to flexibly learn any target cluster distribution depending on the +application domain. +2 + +A PREPRINT - JANUARY 3, 2023 +Factors +Image data +Tabular data +Heterogeneity +Homogeneous pixel distribution +Heterogeneous or multivariate distribution +Spatial Regularity +Yes +No +Sample size +Large, >50,000 +Small, median size ∼ 660 +Benchmark data set +MNIST, CIFAR +No standard benchmark +Data dimensionality +High, >1000 +Low, median 18 +Best method +Deep CNN +Traditional machine learning +Deep approaches +transfer learning, image augmentation +None +Table 1: Contrasts between image and tabular data that require significant rework of deep architectures proposed for +images in learning tabular data. Median sample size and data dimensionality are obtained from 100 most downloaded +tabular data sets from the UCI machine learning repository [25]. +30 +20 +10 +0 +10 +20 +Projected Space 1 +20 +10 +0 +10 +20 +Projected Space 2 +CNV +DME +Drusen +Normal +(a) t-SNE projected +50 +0 +50 +100 +150 +Projected Space 1 +50 +0 +50 +100 +150 +Projected Space 2 +CNV +DME +Drusen +Normal +(b) PCA projected +Figure 1: Two-dimensional embeddings of high dimensional image features extracted from a deep convolutional neural +network obtained from [26]. +3 +Theoretical background +This section provides preliminaries on tabular data in contrast to image data. We draw multiple contrasts between +neighborhood embedding proposed for data visualization and cluster embedding proposed for representation learning +to underpin our proposed approach. +3.1 +Preliminaries +A tabular data set is represented in a matrix X ∈ ℜn×d with n i.i.d samples in rows. Each sample (Xi) is represented +by a d-dimensional feature vector, Xi ∈ ℜd = {x1, x2, . . . , xd}, where i = {1, 2, . . . , n}. Compared to a pixel +distribution P(I) of an image I, tabular data contain multivariate distributions P(x1, x2, . . . , xd) of heterogeneous +variables in relatively much lower dimensions with limited samples. Table 1 shows contrasts between image and +tabular data. One may argue that some high-dimensional sequential data, such as genomics and the MNIST images +converted to pixel vectors, can be structured as tabular data. However, these tabular representations still include +regularity or homogeneity in patterns that do not pose the unique challenges of heterogeneous tabular data. Therefore, +tabular data in business, health records, and many domains fail to take advantage of deep convolutional learning +due to the absence of sequential patterns or image-like spatial regularities. The current literature selectively chooses +data sets with high dimensionality and large sample sizes to take the full benefits of deep learning. In contrast, the +most commonly studied tabular data sets are of low-dimensions and limited samples (Table 1) and are almost never +considered in deep representation learning. Therefore, tabular data sets are identified as the last ”unconquered castle” +for deep learning [15], where traditional machine learning methods are still competing strongly against advanced +neural network architectures [15, 14]. Similar to image learning, there is a need for robust tabular data learning +methods to outperform superior traditional machine learning or clustering methods. +3.2 +Neighborhood embedding +A neighborhood embedding is a low-dimensional map that preserves the similarity between data points (xi and xj) +observed in a higher dimension. Maaten and Hinton propose a Student’s t-distribution to model the similarity between +3 + +A PREPRINT - JANUARY 3, 2023 +t-SNE +DEC [7]/ +[17] +IDEC [8] +Purpose +Neighborhood embedding +Cluster embedding +Low-dimensional +Sampled from Gaussian +Autoencoder +embedding (zi) +with low σ2 +latent space +Distance or similarity +Between sample +Between point & cluster +measure +points (xi, xj) +centroid (xi, µj) +Embedding +t-distribution, +t-distribution, +distribution (qij) +α = 1 +α = 1 +Target +Gaussian in high-dimensional +A function of +distribution (pij) +space (x) +t-distributed qij +Learning +zi+1 = zi + d KLD(p,q) +d(zi) +wi+1 = wi + d KLD(p,q) +d(w) +Purpose +Visualization in d = 2 +Clustering in d > 2 +Table 2: Comparison between neighborhood embedding proposed in t-SNE for data visualization [17] and cluster +embedding proposed in DEC [7] inspired by t-SNE. α = degrees of freedom of t-distribution, d = dimension of low- +dimensional embedding. W represents the trainable parameter of an autoencoder. +samples in neighborhood embedding (zi, zj) of high-dimensional data points (xi and xj) for data visualization [17]. +First, the similarity between two sample points (xi and xj) in the high dimension is modeled by a Gaussian distribution, +pij in Equation 1. Similar joint distribution can be defined for a pair of points in the low-dimensional embedding (zi, +zj) as qij below. +pij = +exp(−||xi − xj||2/2σ2) +� +k̸=l exp(−||xk − xl||2/2σ2), +qij = +exp(−||zi − zj||2/2σ2 +� +k̸=l exp(−||zk − zl||2/2σ2) +(1) +The divergence between the target (pij) and embedding (qij) distributions is measured using a KL divergence loss, +which is minimized to iteratively optimize the neighborhood embedding. +KL (P||Q) = +� +i +� +j +pijlog pij +qij +(2) +To facilitate high-dimensional data visualization in two dimensions (2D), the embedding distribution (qij) is mod- +eled by a Student’s t-distribution, as shown in Equation 3. One primary justification for t-distribution is its heavier +tails compared to a Gaussian distribution. A heavier tail aids in an efficient mapping of outliers observed in high +dimensional space to the 2D space for data visualization. +qij = +(1 + ||zi − zj||)−1 +� +k̸=l(1 + ||zk − zl||)−1 +(3) +Therefore, data points placed at a moderate distance in high-dimension are pulled farther by a t-distribution to aid +visualization in 2D space. In the context of cluster embedding, we argue that the additional separation between points +in low dimensions may alter their cluster assignments. To illustrate this phenomenon, we project high-dimensional +deep convolutional image features on 2D using 1) t-SNE and 2) two principal components, as shown in Figure 1. +The scattering of data points is evident in the t-SNE mapping (Figure 1 (a)), where one blue point appears on the +left side of the figure leading to a wrong cluster assignment, unlike the PCA mapping (Figure 1 (b)). In general, +the expectations of data visualization and clustering tasks are different, as highlighted in Table 2, which should be +considered in respective representation learning. +3.3 +Cluster embedding +Cluster embedding is achieved by infusing cluster separation information into the low-dimensional latent space. While +neighborhood embedding is initialized by sampling from a Gaussian distribution, cluster embedding methods use +embedding learned from an autoencoder’s latent space. However, the current cluster embedding methods use the same +t-distribution (Equation 3) to define the embedding distribution (qij), similar to neighborhood embedding. The target +distribution (pij) is derived as a function of qij, as shown below. +sij = +q2 +ij +� +i qij +, pij = +sij +� +j sij +. +(4) +4 + +A PREPRINT - JANUARY 3, 2023 +While pair-wise sample distances in neighborhood embedding have a complexity of O (N 2), the distances from the +centroids in embedding are O(N*K). Here, K is the number of clusters, which is much smaller than the number +of samples (N). While an outlier point results in N large distances (extremely small pij values) in neighborhood +embedding, there will be much fewer (K<