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+arXiv:2301.02865v1  [astro-ph.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. 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
+[email protected]
+
+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.
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+
+ENERGETIC ELECTRONS AS A DRIVER OF SUNQUAKES
+7
+Table 1. 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.
+

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+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
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+18
+

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+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 : [email protected]
+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.
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+

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+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.
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+preprint arXiv:1909.00429, 2019.
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+Bulletin, 4(3):21–25, 1993.
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+11
+

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+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. The pulsed emission
+from WISEP J101905.63+652954.2 therefore motivates an all-
+sky, untargeted search for pulsed, circularly-polarised emitters
+in LoTTS survey data.
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+Acknowledgements. We thank Dr. Davy Kirkpatrick for making the Keck optical
+spectrum of WISEP J101905.63+652954.2 available to us in machine-readable
+format. 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
+

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: [email protected]
+∗∗e-mail: [email protected]
+∗∗∗e-mail: [email protected]
+
+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
+

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'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content=' acknowledges financial support from FCT (Portugal) under Con-  tract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content=' CEECIND/00488/2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content=' The authors acknowledge the Laboratory for Advanced  Computing at the University of Coimbra (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content='uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content='pt/lca) for providing access to the  HPC resource Navigator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
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+page_content=' Silva, PoS LAT-  TICE2018, 252 (2018), 1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content='11521  [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
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+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
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+page_content=' Parrinello (UKQCD), 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'}
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+page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content='D 61, 079901 (2000)], hep-lat/9811027  [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content=' Aouane, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
+page_content=' Bornyakov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAzT4oBgHgl3EQfSPtl/content/2301.01229v1.pdf'}
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+page_content=' Rev.' 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'}
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+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
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+8
+Fedor V. Fomin, Petr A. Golovach, Jan Kratochvíl, Nicolas Nisse, and Karol Suchan. Pursuing
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+10.1016/j.tcs.2009.12.010.
+9
+Nicolas Nisse and Karol Suchan. Fast Robber in Planar Graphs. In Hajo Broersma, Thomas
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+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
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+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.
+

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+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) = ٠ ��  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 ٠ ��  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 = ٠ ��  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'}
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+page_content=' Fromme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'}
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+page_content='  2  Anthony Bonato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'}
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+page_content='  American Mathematical Society, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE5T4oBgHgl3EQfQg7H/content/2301.05514v1.pdf'}
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+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'}
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+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'}
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+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'}
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+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'}

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+ACCEPTABLE BILATERAL GAMMA PARAMETERS
+YOSHIHIRO SHIRAI
+Department of Mathematics, University of Maryland, College Park
+Abstract. 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: [email protected].
+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
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+
+ACCEPTABLE BILATERAL GAMMA PARAMETERS
+23
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+15, 145–161.
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+arXiv:math/0601035v2 [math.PR].
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+24
+ACCEPTABLE BILATERAL GAMMA PARAMETERS
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+

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+G-CEALS: Gaussian Cluster Embedding in Autoencoder
+Latent Space for Tabular Data Representation
+Manar D. Samad and Sakib Abrar
+Department of Computer Science
+Tennessee State University
+Nashville, TN, USA
+[email protected]
+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<<N) of those large distances in cluster embedding. Therefore, the effect
+of outliers on cluster embedding can be assumed to be much lower compared to the assumption in neighborhood
+embedding.
+4
+Proposed Method
+We propose a novel cluster embedding method, Gaussian Cluster Embedding in Autoencoder Latent Space (G-
+CEALS), by replacing the t-distribution (Equation 3) with a multivariate Gaussian distribution and the target distri-
+bution (Equation 4) with the Gaussian likelihood of individual tabular data samples (Xi) belonging to a given cluster
+(Cj) as P (Xi | Cj) or pij. Two clustering algorithms are used separately in training and evaluating the proposed joint
+cluster embedding method: 1) k-means and 2) Gaussian mixture model (GMM). The clustering on tabular data space
+(Xi ∈ ℜd) yields K cluster assignments for individual samples. Each cluster j is characterized by a centroid vector
+(µj ∈ ℜd) and a covariance matrix (Σj ∈ ℜdxd). Because the dimensionality of tabular data is not as large as image
+data, Σ and µ parameters can be reasonably sized for computation. Therefore, the soft cluster assignment (Sx(i, j))
+for individual samples can be obtained using a Gaussian kernel, which is the negative exponent of the Mahalanobis
+distance (dx(i, j)) between the point (Xi) and the j-th cluster centroid vector, as shown in Equations 5 and 6.
+dx(i, j)
+=
+�
+(Xi − µx
+j ) Σ−1
+x
+(Xi − µj)T
+(5)
+Sx(i, j)
+=
+exp (−d2
+x(i, j))
+(6)
+To ensure that the sum of all soft cluster assignments equals one for a given sample, we obtain a joint cluster distribu-
+tion, P (xi, µj) or pij, as shown in Equation 7. We set pij, obtained via superior traditional machine learning, as the
+target distribution to improve the autoencoder embedding (zi) of tabular data. Similarly, the embedding distribution
+(qij) can be obtained using soft Gaussian cluster assignments (Sz) on the low-dimensional latent space (zi), as shown
+in Equation 7.
+pij =
+Sx(i, j)
+�
+j Sx(i, j),
+qij =
+Sz(i, j)
+�
+j Sz(i, j)
+(7)
+Therefore, our P and Q distributions are independent, unlike the current cluster embedding methods. Additionally,
+the covariance of the target (Σx) and embedding (Σz) distributions can regulate the scatter or compactness of data
+clusters, which is impossible with t-distributed embedding.
+Algorithm 1 Proposed G-CEALS Algorithm
+Input: d-dimensional tabular data, X ∈ ℜn×d, where Xi ∈ ℜd
+Output: Tabular data embedding, Z ∈ ℜn×m, m<<d
+j − th cluster parameters, {µx
+j , Σx
+j } ← K-means or GMM clustering of X
+pij ← {µx
+j , Σx
+j }, in Equation 7
+Initialize: W 0 ={Wencoder, Wdecoder}
+for t = 1 → n epochs do
+{ ˆX, Zt} ← Encoder (X, W t
+encoder)
+{µz
+j, qij} ← K-means or GMM clustering of Zt and using qij in Equation 7
+L← Lauto + γ ∗ Lcluster, measure the loss terms in Equation 9
+W t ← AutoEncoder (W t−1), update weights minimizing the joint loss in Equation 9
+end for
+4.1
+Low-dimensional embedding optimization
+A single-layer autoencoder is trained to encode the input (X) to a latent space (Z), which is then decoded to reconstruct
+the original input ( ˆ
+Xi), as shown in Equation 8.
+Lauto = argmin
+θ,Φ
+N
+�
+i=1
+||Xi − ˆ
+Xi||2
+2.
+(8)
+5
+
+A PREPRINT - JANUARY 3, 2023
+Data set
+Sample size
+Dimensions
+Classes
+Domain
+Breast Cancer
+569
+30
+2
+Diagnostic
+Dermatology
+358
+34
+6
+Histopathological
+E. coli
+336
+7
+8
+Protein cell
+TUANDROMD
+4465
+241
+2
+Android malware
+Mice Protein
+552
+78
+8
+Protein expression
+Olive
+572
+10
+3
+Food & beverage
+Vehicle
+846
+18
+4
+Silhouette features
+Table 3: Summary of tabular data sets used for comparing clustering performance
+Here, θ and Φ denote the trainable parameters of the encoder and decoder, respectively. The embedding obtained
+following each epoch of training is clustered using a clustering algorithm to obtain the cluster parameters (µ, Σ) and
+the embedding distribution (qij). Given the target distribution (pij) (Equation 7), one of the learning objectives of
+G-CEALS is to minimize the KL divergence between P and Q distributions (Lcluster), as shown in Equation 9 below.
+The overall learning objective of G-CEALS is to update the autoencoder’s weights by minimizing a joint cost function,
+the encoder reconstruction loss and the KL divergence loss, as below.
+L
+=
+Lauto + γ ∗ Lcluster
+=
+argmin
+θ,Φ
+N
+�
+i=1
+||Xi − ˆ
+Xi||2
+2 + γ ∗
+N
+�
+i=1
+K
+�
+j=1
+pijlog pij
+qij
+(9)
+Here, γ is a trade-off hyperparameter to balance the contribution of cluster divergence Lcluster during representation
+learning. The G-CEALS algorithm is summarized in Algorithm 1.
+5
+Experiments
+All experimental steps and algorithms are implemented and evaluated in Python. The neural networks are built using
+the PyTorch package, and clustering modules are developed using the sci-kit-learn package1 We evaluate the proposed
+and baseline methods on seven multi-domain and multivariate tabular data sets. A summary of these tabular data sets
+is provided in Table 3.
+5.1
+Adapting image learning architectures to tabular data
+A concurrent study compares the baseline embedding clustering methods on tabular data sets [27]. The comparison
+results reveal that state-of-the-art embedding clustering methods proposed for image data may not be optimal baselines
+for non-image tabular data. For example, Caron et al. have learned visual features from images using AlexNet and
+VGG-16 after Sobel filtering for color removal and contrast enhancement, which do not apply to tabular data [6]. Their
+deepCluster architecture has five convolutional layers with up to 384 2D image filters to learn image texture. Transfer
+learning of tabular data, similar to VGG-16 on images, is not intuitive because tabular datasets do not share transferable
+textures. Furthermore, these deep architectures can easily overfit data with limited sample size and dimensionality,
+similar to tabular data sets. The DEPICT method uses a convolutional denoising autoencoder for reconstructing
+original images from corrupted images [20]. Because similar image corruption is not trivial on data tables, we use
+standard CNN autoencoders with single and three layers as baseline methods. The deep clustering network (DCN)
+method uses a fully-connected deep neural network (FC-DNN) with 2000, 1000, 1000, 1000, and 50 neurons for
+learning high-dimensional image data [11], whereas tabular data can have as low as ten input features. They avoid
+CNN architecture to focus on their DCN learning objective instead of exhaustively searching all learning architectures.
+However, they leverage a stacked deep autoencoder architecture instead of using a regular autoencoder model, which
+may overshadow the original contribution of the algorithm. Similarly, the deep k-means (DKM) method used FC-
+DNN instead of CNN for image learning [9]. The DKM method is compared against the ones that use FC-DNN
+(excluding all CNN-based methods) to avoid architectural bias. We use the original DKM method to reproduce cluster
+embedding on tabular data. Recently, Mrabah et al., in their Dynamic Autoencoder (DynAE) method, have used image
+augmentation (shifting and rotation), which is not intuitive with tabular data, and a 2D convolutional adversarial
+autoencoder for image data, which needs substantial customization of the adversarial network for learning feature
+vectors in tabular format [10]. One limitation of the DynAE and DKM methods is that the latent dimension is restricted
+to the number of clusters, whereas our method is proposed for any latent dimension.
+1The source code will be shared publicly and kept private for anonymity during peer review.
+6
+
+A PREPRINT - JANUARY 3, 2023
+1
+2
+3
+4
+5
+Data 
+folds
+Trained model 
+with best ?
+? = [0.1, 0.2, 0.3, .... 1.0]
+Accuracy
+NMI
+Hyperparameter 
+search
+(p,q)
+Z
+x
+x?
+Test fold
+....... .
+.
+. ... ... ..
+Clustering
+Z
+Figure 2: Five-fold cross-validation scheme for training and tuning the model with the best γ value (Equation 9). The
+clustering accuracy is reported on the embedding of the left-out test data fold.
+5.2
+Baseline methods
+The comparison of six state-of-the-art cluster embedding methods on tabular data sets reveals the IDEC method as the
+best performing, followed by traditional clustering (k-means and GMM) as the second best competitive baseline [27].
+Furthermore, we detail in the previous section why existing methods for image learning may not be appropriate
+baselines for tabular data sets without some customization. The limited sample size and dimensionality of tabular
+data may require a simpler learning architecture than image data. For example, the latent space size is set to 256 for
+image learning, whereas the input dimension of tabular data can be as low as 10.
+Considering these factors, we compare our proposed method against four competitive baseline methods. First, the
+k-means and GMM clustering are performed on input tabular data (X) because traditional machine learning methods
+are known to produce competitive results on tabular data, unlike image data. Second, a two-stage method is used:
+1) the embedding (Z) extraction by training a single-layer autoencoder and then 2) perform clustering on Z [28,
+29]. Third, embedding learning and clustering are performed jointly on tabular data. We compare fully-connected
+autoencoder (FC-AE) and CNN autoencoder (similar to the DEPICT method [20]) in single-layer and three-layer
+settings to investigate a suitable learning architecture for tabular data. Fourth, despite the challenges in adapting
+image learning methods to tabular data learning as mentioned in the previous section, we use two pioneering methods
+for cluster embedding, DEC [7], IDEC [8], and a more recent method (deep k-means) DKM [9] for tabular data
+to compare with our proposed method. Although sophisticated autoencoder architectures proposed in the literature
+(stacked, variational, adversarial, convolutional) can be compared in an exhaustive search for the best method, it will
+introduce architectural bias in our claim for the best learning algorithm. Therefore, we use a single-layer autoencoder
+to make a fair comparison between our and the baseline learning algorithms, especially considering the size and
+dimensionality of tabular data.
+5.3
+Evaluation
+The proposed G-CEALS model training involves self-supervised data reconstruction and unsupervised clustering with-
+out requiring any ground truth. However, existing studies show that the hyperparameter (γ in Equation 9) value is
+data-dependent and is tuned based on clustering accuracy that requires ground truth labels. The quality of cluster
+embedding is evaluated in downstream clustering tasks using two standard metrics: clustering accuracy (ACC) [30]
+and normalized mutual information (NMI) [31]. We follow the same metrics for hyperparameter tuning and cluster
+embedding evaluation. However, existing methods report the clustering accuracy on the same training data set because
+of the unsupervised nature of the problem. In contrast, we use a semi-supervised five-fold cross-validation scheme to
+obtain reproducible and transferable learning for downstream clustering or classification. As shown in Figure 2, four
+data folds are used in unsupervised training and supervised tuning, which is then used to obtain the embedding of a
+left-out test data fold. We report the average ACC and NMI scores across the five left-out data folds to compare the
+proposed and baseline methods. These metrics score between 0 (failure) and 1 (perfect clusters). For all evaluation
+purposes, the cluster number is set to the number of class labels for a given data set. The scores are multiplied by 100
+to represent the numbers in percentage.
+7
+
+A PREPRINT - JANUARY 3, 2023
+Data set
+FC-AE
+FC-AE
+CNN
+CNN
+CNN
+FC-AE
+GMM
+K-means
+GMM
+K-means
+GMM
+GMM
+NHL
+1
+1
+1
+1
+3
+3
+Breast cancer
+ACC
+91.2 (4.8)
+85.8 (3.3)
+92.6 (3.3)
+89.8 (4.3)
+62.7 (3.1)
+85.4 (12.0)
+NMI
+59.2 (18.1)
+43.8 (10.6)
+63.8 (11.7)
+55.4 (11.2)
+0.0 (0.0)
+49.4 (24.0)
+Dermatology
+ACC
+77.2 (9.8)
+76.4 (10.1)
+75.7 (5.4)
+72.3 (4.9)
+31.0 (3.8)
+74.3 (6.6)
+NMI
+77.6 (5.8)
+77.4 (5.4)
+77.8 (4.5)
+76.1 (4.5)
+0.0 (0.0)
+82.9 (5.0)
+E. coli
+ACC
+32.2 (3.7)
+31.6 (3.7)
+27.7 (2.3)
+29.2 (2.5)
+34.2 (4.6)
+32.4 (1.5)
+NMI
+18.0 (6.8)
+17.4 (3.1)
+17.7 (4.2)
+17.1 (3.7)
+13.3 (3.3)
+15.6 (3.2)
+TUANDROMD
+ACC
+83.4 (6.6)
+40.8 (4.0)
+79.4 (1.6)
+79.6 (1.2)
+79.4 (1.6)
+80.6 (4.5)
+NMI
+18.8 (23.8)
+36.0 (3.3)
+0.4 (0.2)
+2.3 (1.0)
+0.4 (0.2)
+7.9 (14.6)
+Mice protein
+ACC
+42.0 (1.8)
+40.6 (1.7)
+37.7 (2.2)
+36.2 (3.0)
+18.8 (2.4)
+39.9 (2.1)
+NMI
+40.4 (3.0)
+38.0 (3.9)
+36.4 (1.3)
+33.3 (2.7)
+0.0 (0.0)
+40.8 (3.3)
+Olive
+ACC
+70.8 (7.9)
+73.6 (6.4)
+58.9 (5.5)
+71.2 (10.6)
+56.5 (6.1)
+66.8 (8.7)
+NMI
+42.6 (9.0)
+47.0 (6.6)
+30.7 (6.9)
+44.4 (10.2)
+0.0 (0.0)
+39.8 (10.4)
+Vehicle
+ACC
+41.2 (3.3)
+44.2 (3.5)
+40.8 (2.2)
+42.4 (4.6)
+40.7 (4.3)
+42.1 (4.2)
+NMI
+11.8 (2.8)
+14.6 (2.2)
+11.5 (2.3)
+13.5 (4.9)
+12.7 (1.9)
+14.8 (3.4)
+Table 4: Comparing fully-connected autoencoder (FC-AE) with CNN-autoencoder for the proposed G-CEALS algo-
+rithm in single or three-layer settings. NHL = Number of hidden or convolutional layers.
+6
+Results
+All experiments are conducted on a Dell Precision 5820 workstation running Ubuntu 20.04 with 64GB RAM and an
+NVIDIA GeForce RTX 3080 GPU with 10GB memory. We standardize all tabular data using the mean and standard
+deviation of individual variables before training the autoencoder or performing clustering.
+6.1
+Learning architecture and model selection
+We compare the performance of fully connected autoencoders (FC-AE) and convolutional autoencoders (CNN-AE)
+in single- and three-layer architectures for our tabular data sets. Table 4 clearly shows the superiority of single-
+layer FC-AE architecture over CNN and deeper architectures, which we select in our subsequent analysis. A single-
+layer autoencoder maps a d-dimensional tabular data sample to a five-dimensional autoencoder latent space (d>5),
+considering the range of dimensionality of our tabular data sets (seven to 241). For all experiments, the learning rate is
+set to 0.0001 with an Adam optimizer. The best autoencoder model jointly trained clustering is selected by searching
+the best epoch point and γ value while training it for a maximum of 5000 epochs. The best gamma value is searched
+from a range between 0.1 and 1.0.
+6.2
+Clustering of tabular data versus latent space
+Tables 5 and 6 show clustering scores and rank ordering for nine methods, respectively. Traditional clustering (K-
+means and GMM) on tabular data yields the top three scores for the dermatology, breast cancer, mice protein, and
+olive data sets. This finding is at odds with the previous finding that direct clustering of images in pixel space yields
+the worst performance. This is because tabular data sets have relatively lower dimensionality and the absence of
+regularity in patterns makes such data still suitable for traditional machine learning. For example, these clustering
+methods yield the worst (<1.0, max. 100) NMI scores for the highest dimensional (241) TUANDROMD data set.
+Alternatively, clustering methods (GMM, K-means) can be applied to the autoencoder’s latent space (Z). A trained
+autoencoder is used to obtain the embedding on test data folds. The test data embedding is then clustered using GMM
+and K-means, which are presented as GMM on Z and K-means on Z in Table 5, respectively. Except for the E. coli
+data set, GMM on Z performs worse than GMM clustering of other data sets. Similarly, K-means clustering of tabular
+data yields substantially better accuracy than K-means clustering of Z, except for the vehicle data set.
+6.3
+Clustering of joint cluster embedding
+The autoencoder latent space Z is jointly learned with data cluster distributions in this method. Our results on tabular
+data are reproduced using two pioneering cluster embedding methods: DEC [7] and IDEC [8]. The DEC method
+appears to be among the worst of nine methods presented in Table 5, except for the vehicle data set. Therefore, a
+method proposed for image learning may not perform equally well on tabular data. However, the improved DEC
+8
+
+A PREPRINT - JANUARY 3, 2023
+Data set
+GMM
+K-means
+GMM
+K-means
+DEC
+IDEC
+DKM
+G-CEALS
+G-CEALS
+on X
+on X
+on Z
+on Z
+GMM
+K-means
+Breast cancer
+ACC
+89.8 (4.6)
+90.2 (4.3)
+82.2 (7.4)
+82.8 (5.7)
+68.0 (3.0)
+86.0 (3.6)
+64.2 (3.9)
+91.2 (4.8)
+85.8 (3.3)
+NMI
+55.4 (17.6)
+56.2 (17.0)
+40.6 (19.6)
+39.2 (15.9)
+9.2 (6.2)
+44.4 (11.3)
+2.7 (7.2)
+59.2 (18.1)
+43.8 (10.6)
+Dermatology
+ACC
+76.8 (8.8)
+76.2 (9.2)
+72.2 (8.8)
+63.0 (4.7)
+50.4 (7.4)
+76.6 (12.2)
+23.2 (0.5)
+77.2 (9.8)
+76.4 (10.1)
+NMI
+82.4 (4.2)
+83.4 (5.0)
+73.4 (5.0)
+70.8 (4.4)
+45.2 (6.9)
+80.6 (7.2)
+3.5 (0.3)
+77.6 (5.8)
+77.4 (5.4)
+Ecoli
+ACC
+29.4 (4.5)
+29.2 (3.2)
+30.6 (4.7)
+29.0 (4.0)
+26.2 (2.1)
+32.6 (3.7)
+35.4 (2.9)
+32.2 (3.7)
+31.6 (3.7)
+NMI
+19.6 (6.6)
+18.4 (5.6)
+17.8 (6.5)
+18.6 (6.3)
+15.0 (5.3)
+17.4 (6.3)
+14.1 (2.7)
+18.0 (6.8)
+17.4 (3.1)
+TUANDROMD
+ACC
+79.1 (1.7)
+79.1 (1.7)
+77.4 (2.6)
+77.2 (3.6)
+79.2 (1.5)
+82.0 (4.1)
+48.7 (11.3)
+83.4 (6.6)
+40.8 (4.0)
+NMI
+0.5 (0.2)
+0.6 (0.1)
+1.6 (2.1)
+0.5 (0.2)
+0.8 (0.7)
+13.8 (9.7)
+6.8 (5.6)
+18.8 (23.8)
+36.0 (3.3)
+Mice protein
+ACC
+40.8 (1.7)
+40.2 (5.7)
+36.4 (5.1)
+35.0 (2.5)
+34.8 (3.1)
+35.6 (2.3)
+17.2 (2.7)
+42.0 (1.8)
+40.6 (1.7)
+NMI
+42.0 (3.2)
+40.6 (4.5)
+31.6 (3.8)
+34.2 (4.2)
+30.8 (6.2)
+35.6 (3.6)
+3.1 (9.1)
+40.4 (3.0)
+38.0 (3.9)
+Olive
+ACC
+67.2 (11.5)
+71.4 (10.8)
+57.6 (11.2)
+59.4 (12.7)
+55.8 (5.5)
+77.2 (4.0)
+56.5 (0.0)
+70.8 (7.9)
+73.6 (6.4)
+NMI
+39.4 (12.6)
+43.2 (11.7)
+30.0 (13.2)
+32.6 (14.0)
+25.4 (5.5)
+49.4 (5.2)
+0.0 (0.0)
+42.6 (9.0)
+47.0 (6.6)
+Vehicle
+ACC
+39.4 (2.9)
+37.2 (1.7)
+39.0 (2.6)
+39.0 (2.8)
+41.4 (3.3)
+42.8 (3.4)
+31.1 (5.6)
+41.2 (3.3)
+44.2 (3.5)
+NMI
+13.6 (1.0)
+12.4 (1.6)
+13.2 (1.5)
+13.0 (1.7)
+12.8 (1.5)
+17.6 (3.8)
+6.4 (6.3)
+11.8 (2.8)
+14.6 (2.2)
+Table 5: Clustering accuracy (ACC) and normalized mutual information (NMI) scores of proposed and baseline meth-
+ods. Z = autoencoder latent space without joint learning. X = tabular data space. The DKM method is used on
+tabular data set without customizing this image learning method for tabular data. Otherwise, a single-layer autoen-
+coder without pre-training is used for all representation learning methods to avoid architectural bias in comparing the
+algorithms.
+Data set
+GMM
+K-means
+GMM
+K-means
+DEC
+IDEC
+DKM
+CNN
+Proposed
+on X
+on X
+on Z
+on Z
+GMM
+G-CEALS
+Breast cancer
+4
+3
+7
+6
+8
+5
+9
+1
+2
+Dermatology
+2
+4
+6
+7
+8
+3
+9
+5
+1
+E. coli
+5
+6
+4
+7
+9
+2
+1
+8
+3
+TUANDROMD
+5
+6
+7
+8
+4
+2
+9
+3
+1
+Mice protein
+2
+3
+5
+7
+8
+6
+9
+4
+1
+Olive
+4
+2
+7
+5
+9
+1
+8
+6
+3
+Vehicle
+5
+8
+6
+7
+3
+2
+9
+4
+1
+Average
+3.9 (1.3)
+4.6 (2.0)
+6.0 (1.1)
+6.7 (0.9)
+7.0 (2.3)
+3.0 (1.7)
+7.7 (2.8)
+4.4 (2.1)
+1.4 (0.7)
+Table 6: Data set-specific and average ranks of the proposed and baseline methods based on clustering accuracy.
+CNN-GMM is a single convolutional layer autoencoder trained jointly with GMM via the proposed algorithm. The
+DKM method is used without any customization or pretraining.
+(IDEC) method shows substantial improvement over the DEC method. The IDEC method always outperforms the
+GMM on Z except for the mice protein data set. K-means clustering on Z is always inferior to the IDEC method.
+The IDEC is the best of all methods for the olive data sets. Unlike DEC and IDEC methods, we do not modify the
+DKM method for the tabular data sets. Using the original image learning architecture (500-500-2000 neurons) on
+tabular data, the DKM method yields the worst of all clustering accuracy with a zero NMI score for the olive data set.
+However, it performs the best for the Ecoli data set with the lowest dimensionality (8) of all data sets.
+6.4
+Proposed G-CEALS method
+Table 5 shows that our proposed G-CEALS method jointly trained with GMM clustering (ACC: 91.2) outperforms
+the second-best method, K-means clustering (ACC: 90.2), for the breast cancer data set. The proposed G-CEALS
+method with GMM clustering (ACC: 77.2) also outperforms the second-best method, GMM clustering (ACC: 76.8)
+for the dermatology data set. For the E. coli data set, the proposed method (ACC: 32.2) is on par with the second-
+best method, IDEC (ACC: 32.6). The G-CEALS method (ACC: 83.4) outperforms the second-best IDEC (ACC:82.0)
+method on the TUANDROMD data set. The proposed method with GMM clustering (ACC: 42.0) again outperforms
+the second-best method, GMM clustering (ACC: 40.8), on the mice protein data set. Only for the Olive data set,
+the IDEC method (ACC: 77.2) outperforms the proposed G-CEALS method with K-means clustering (ACC: 73.6).
+However, the G-CEALS method with K-means clustering (ACC: 44.2) outperforms the second best method, IDEC
+(ACC: 42.8) on the vehicle data set. Table 6 shows that the proposed G-CEALS method ranks the best method on
+four of the seven data sets. For the other three data sets, G-CEALS ranks among the top three. The average ranking
+reveals that our method (average rank: 1.4) substantially outperforms the second-best (IDEC, average rank: 3.0) and
+the third-best method: GMM clustering of tabular data (average rank: 3.9).
+9
+
+A PREPRINT - JANUARY 3, 2023
+7
+Discussion of results
+The paper is one of the first studies to jointly learn embedding and clustering in an autoencoder latent space with
+tabular data. The findings of this article can be summarized as follows. First, traditional clustering on tabular data
+is competitive with clustering on the autoencoder latent space. Our method outperforms these superior methods for
+clustering. Second, joint embedding learning with a cluster distribution (IDEC and our method) shows improved data
+representation over all disjoint learning methods (DEC, K-means on Z, GMM on Z) and GMM/K-means clustering
+on tabular data. Including the DKM method, cluster embedding methods yield the best performance on all seven
+tabular data sets. Third, our assumption of Gaussian clusters and an independent target distribution have improved the
+clustering accuracy over the methods using t-distributed clusters and all other baselines on average. We elaborate on
+these findings in the following sections.
+7.1
+Image versus tabular data embedding
+In computer vision, clustering images on high-dimensional image pixel space is ineffective, as reported in previous
+studies. On the other hand, deep learning methods simultaneously and efficiently achieve dimensionality reduction,
+feature learning via convolution operations, and classification in an end-to-end framework. In contrast, tabular data
+sets appear with relatively smaller sample sizes and dimensionality than image data and do not contain regularity in
+features to leverage the benefit of convolutional feature extraction. Our results confirm that deep models with three
+hidden layers or convolutional networks or clustering on autoencoder embedding are not effective for tabular data.
+This finding confirms the superiority of traditional machine learning on tabular data over deep learning. However, our
+proposed representation learning method outperforms the superior clustering method to demonstrate that tabular data
+need special algorithms to perform effectively with neural network-based learning architectures. G-CEALS performs
+good with the GMM clustering algorithm may be due to the multivariate Gaussian modeling of the embedding distri-
+butions. Clustering on autoencoder embedding (Z) is unsatisfactory because the reconstruction loss may have altered
+the cluster distribution in the latent space. Therefore, setting the superior cluster distribution on tabular data space
+as the target may have helped in improving the quality of the cluster embedding. The G-CEALS method is designed
+for tabular data and may not be appropriate for images because the target distribution would be ineffective on pixel
+space. Overall, the cluster embedding of tabular data requires a learning algorithm and architecture distinct from those
+proposed for image data.
+7.2
+Trade-off between data reconstruction and cluster divergence loss
+We observe that one of the first successful deep cluster embedding methods proposed for image clustering (DEC [7])
+performs the worst among all approaches with tabular data sets 6. The DEC method first pretrains an autoencoder
+and then separately finetunes the KL divergence loss after modeling a t-distributed embedding for clustering. There is
+no γ parameter because the Lauto and Lcluster optimizations happen separately in DEC. Conversely, the introduction
+of γ parameter in IDEC and our method has substantially improved the quality of cluster embedding. While the
+autoencoder reconstruction loss Lauto retains all information in the latent space, Lcluster (Equation 9) disrupts the
+latent space to learn the clusters. A disproportionate contribution of these two loss terms can collapse the cluster
+distribution in the embedding leading to poor cluster performance. Therefore, tuning the γ parameter is a crucial step
+in cluster embedding methods.
+7.3
+G-CEALS versus other methods
+Our results show that the IDEC method performs better in cases where the clustering on tabular data space is sub-
+stantially worse (E. coli and Olive data sets in Table 6). It may be because the IDEC method does not learn from
+the cluster distribution on the tabular data space, similarly to the proposed G-CEALS method. In IDEC, the target
+distribution is derived from the t-distributed embedding instead. Overall, finding an appropriate target distribution for
+optimizing the KL divergence loss remains an open problem for future work. Another common observation is that
+the Olive and E. coli data sets have the lowest dimensionality among the seven tabular data sets (Table 3) with only
+ten and seven variables, respectively. Therefore, it may be inferred that the IDEC method is superior to the proposed
+G-CEALS method when the dimensionality of tabular data is less than ten. For tabular data sets with the highest
+dimensionality ( TUANDROMD (241) and mice protein (78)), the G-CEALS method outperforms the IDEC method
+as the best-performing method. This observation suggests that Gaussian embedding may be more suitable over its
+t-distributed counterparts when a tabular data set has higher dimensionality.
+10
+
+A PREPRINT - JANUARY 3, 2023
+7.4
+Limitations
+While tunable parameters provide sufficient flexibility to optimize the embedding, finding the best parameter setting
+can be computationally expensive and data-demanding. The quality of embedding can be sensitive to the choice of
+these hyperparameters due to the lack of large training samples. The G-CEALS method in this paper uses the clustering
+accuracy (ACC) metric to search and select the trained model for embedding generation. This may help the model
+obtain the best ACC scores on test data embedding. As mentioned earlier, the G-CEALS method is not suitable for
+image data because the target distribution is obtained in the high-dimensional input feature space.
+8
+Conclusions
+This paper presents a novel cluster embedding method in one of the first studies on tabular data sets. The superiority
+of our G-CEALS method suggests that multivariate Gaussian distribution is superior to the widely used t-distribution
+assumption for learning tabular data embedding. Our findings show that tabular data sets require learning algorithms
+and architectures distinct from those proposed for image learning. This data-centric learning approach may improve a
+deep model’s performance on tabular data over its currently known superior machine learning counterparts. The pro-
+posed joint learning framework provides a promising representation learning of tabular data over its superior machine
+learning counterparts.
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