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+Astronomy & Astrophysics manuscript no. core
+©ESO 2023
+January 30, 2023
+Expanding Sgr A* dynamical imaging capabilities with an African
+extension to the Event Horizon Telescope
+Noemi La Bella1, Sara Issaoun2, 3, 1, Freek Roelofs2, 4, 1, Christian Fromm5, 6, 7, and Heino Falcke1
+1 Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box
+9010, 6500 GL Nijmegen, The Netherlands
+e-mail: [email protected]
+2 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
+3 NASA Hubble Fellowship Program, Einstein Fellow
+4 Black Hole Initiative, Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
+5 Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Emil-Fischer-Strasse 31, 97074 Würzburg, Germany
+6 Institut für Theoretische Physik, Goethe Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt, Germany
+7 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
+January 30, 2023
+ABSTRACT
+Context. The Event Horizon Telescope (EHT) has recently published the first images of the supermassive black hole at the center of
+our Galaxy, Sagittarius A* (Sgr A*). Imaging Sgr A* is plagued by two major challenges: variability on short (approximately minutes)
+timescales and interstellar scattering along our line of sight. While the scattering is well studied, the source variability continues to
+push the limits of current imaging algorithms. In particular, movie reconstructions are hindered by the sparse and time-variable
+coverage of the array.
+Aims. In this paper, we study the impact of the planned Africa Millimetre Telescope (AMT, in Namibia) and Canary Islands telescope
+(CNI) additions to the time-dependent coverage and imaging fidelity of the EHT array. This African array addition to the EHT further
+increases the eastwest (u, v) coverage and provides a wider time window to perform high-fidelity movie reconstructions of Sgr A*.
+Methods. We generated synthetic observations of Sgr A*’s accretion flow and used dynamical imaging techniques to create movie
+reconstructions of the source. To test the fidelity of our results, we used one general-relativistic magneto-hydrodynamic model of the
+accretion flow and jet to represent the quiescent state and one semi-analytic model of an orbiting hotspot to represent the flaring state.
+Results. We found that the addition of the AMT alone offers a significant increase in the (u, v) coverage, leading to robust averaged
+images during the first hours of the observating track. Moreover, we show that the combination of two telescopes on the African
+continent, in Namibia and in the Canary Islands, produces a very sensitive array to reconstruct the variability of Sgr A* on horizon
+scales.
+Conclusions. We conclude that the African expansion to the EHT increases the fidelity of high-resolution movie reconstructions of
+Sgr A* to study gas dynamics near the event horizon.
+Key words. Black hole physics - Galaxy: center - Instrumentation: high angular resolution - interferometers - Techniques: image
+processing
+1. Introduction
+The Event Horizon Telescope (EHT) collaboration has recently
+published the first images of the black hole shadow of Sagittar-
+ius A* (Sgr A*), the supermassive black hole (SMBH) at the
+center of the Milky Way, characterized by an asymmetric bright
+ring of (52.1 ± 0.6) µas (Event Horizon Telescope Collaboration
+et al. 2022a). The ring-like morphology was recovered in over
+95% of the best-fit images produced from 2017 April 6 and 7
+observations. The EHT images of Sgr A* are consistent with the
+prediction of a shadow for a Kerr black hole (Falcke et al. 2000)
+with a mass M ∼ 4 × 106M⊙ at a distance D ∼ 8 kpc, which
+were accurately measured by high-resolution infrared studies of
+stellar orbits in the Galactic Center (Gravity Collaboration et al.
+2018; Do et al. 2019). In 2019, the EHT collaboration delivered
+the first ever image of a black hole shadow in the giant ellipti-
+cal galaxy M87 (Event Horizon Telescope Collaboration et al.
+2019a). The main difference between the two SMBHs is their
+mass. M87* is about 1600 times more massive than Sgr A* and
+thus, it has a longer gravitational timescale. In fact, the period of
+the innermost stable circular orbit (ISCO) for a nonrotating black
+hole as massive as M87* is ∼30 days, while for Sgr A* it is ∼30
+minutes. As a consequence, the estimation of the ring diameter
+of Sgr A* is more uncertain than in M87* and we need movies
+to properly study the plasma motion surrounding the black hole
+on this short orbital timescale.
+The variability of Sgr A* required a reformulation of the
+static source assumptions in the interferometric Earth aperture
+synthesis method and imaging algorithms used for M87*. In par-
+ticular, to generate a typical static image of Sgr A*, a variabil-
+ity noise budget needs to be added, while a dynamical imaging
+process is required to capture the evolving structure of Sgr A*
+(Event Horizon Telescope Collaboration et al. 2022b). Because
+of the sparsity of the EHT array, time slots with good (u, v) cover-
+age were selected to perform dynamical studies on the variability
+(Farah et al. 2022).
+Article number, page 1 of 11
+arXiv:2301.11384v1  [astro-ph.IM]  26 Jan 2023
+
+A&A proofs: manuscript no. core
+The SMBH also presents flare events observed across the
+electromagnetic spectrum in the last decades. An accurate study
+of the millimeter light curves during the 2017 EHT campaign
+was done by Wielgus et al. (2022a). In particular, the authors
+found excess variability on 2017 April 11, following a flare
+observed in the X-ray. Subsequent studies on polarized light
+curves with the Atacama Large Millimeter/submillimeter Array
+(ALMA) on the same day (Wielgus et al. 2022b) revealed the
+presence of a hotspot orbiting Sgr A* clockwise.
+In addition to its quiescent variability, imaging Sgr A* is a
+complex process because the very long baseline interferometry
+(VLBI) observations are affected by scattering in the interstel-
+lar medium along our line of sight toward the Galactic Center.
+The consequent diffractive and refractive effects of the scatter-
+ing were mitigated by modeling their chromatic properties in the
+radio band (see Psaltis et al. 2018; Johnson et al. 2018; Issaoun
+et al. 2019a, 2021; Event Horizon Telescope Collaboration et al.
+2022b, for more details).
+Eight telescopes at six geographic locations formed the 2017
+EHT array configuration that led to the first images of Sgr A*
+and M87*. Since 2017, the array has doubled in bandwidth and
+increased the number of baselines with three new telescopes.
+As of 2022, the EHT has consisted of eleven telescopes at
+eight locations: ALMA and the Atacama Pathfinder Experiment
+(APEX) on the Llano de Chajnantor in Chile; the Large Millime-
+ter Telescope (LMT) Alfonso Serrano on the Volcán Sierra Ne-
+gra in Mexico; the James Clerk Maxwell Telescope (JCMT) and
+Submillimeter Array (SMA) on Maunakea in Hawai’i; the In-
+stitut de Radioastronomie Millimétrique 30-m telescope on Pico
+Veleta (PV) in Spain; the Submillimeter Telescope (SMT) on Mt.
+Graham and the 12-m telescope on Kitt Peak (KP) in Arizona;
+the South Pole Telescope (SPT) in Antarctica; the Northern Ex-
+tended Millimeter Array (NOEMA) in France; and the Green-
+land Telescope (GLT) at Thule. This new configuration offers
+increased sensitivity of the array and will enable higher-fidelity
+images of Sgr A* and M87*. However, all new telescopes are
+in the northern hemisphere and are less effective for imaging
+southern sources. Additional telescopes are being considered to
+expand the capabilities of the array, especially on the African
+continent, which offers prime site locations to increase the (u, v)
+coverage toward Sgr A*.
+In this work, we consider two additions to the EHT in the
+African region: one in Namibia and one in the Canary Islands.
+The Africa Millimetre Telescope (AMT), planned on Mt. Gams-
+berg (2,347 m a.s.l.) in Namibia, will be the first millimeter-wave
+telescope in Africa. The project to add this telescope is currently
+underway, and aims to relocate the decommissioned 15-meter
+SEST telescope in Chile to Gamsberg in the next years. This site
+will offer low precipitable water vapor levels during the typical
+fall and spring EHT campaign seasons (Backes et al. 2016) and
+its strategical position in the southern hemisphere at the same lat-
+itude as ALMA provides important eastwest baselines to Chile
+and northsouth baselines to Europe, significantly increasing the
+snapshot coverage in the first half of a typical observing night.
+The island of La Palma in the Canary Islands (2,000 m a.s.l.)
+has dry weather conditions throughout the year (Raymond et al.
+2021) and offers a prime location to provide mid-range coverage
+between Namibia and Europe that is crucial to constrain source
+compactness and extent. Furthermore, the site’s established in-
+frastructure from existing observatories would make an addi-
+tional telescope easily feasible and well supported, making it an
+ideal candidate for a telescope location in the near term.
+We present simulated dynamical images of Sgr A* using
+the 2022EHT array and an African extension including two new
+telescopes: the 15-meter AMT, and the Canary Islands telescope,
+CNI, on the island of La Palma. We assume the dish size of
+CNI to be six meters, following the design concept for a next-
+generation EHT facility in the long term (Doeleman et al. 2019).
+We investigate the impact of the AMT and CNI stations on imag-
+ing Sgr A* in both quiescent and flaring states. The methods we
+use can easily be expanded to other EHT configurations.
+The paper is organized as follows: in Section 2, we describe
+the synthetic generation pipeline and imaging algorithms used.
+In Section 3, we present the African extension to the EHT and
+its contribution to snapshot and full-track (u, v) coverage. In Sec-
+tion 4, we show the static and dynamical reconstructions ob-
+tained with the enhanced EHT array. Finally, in Section 5, we
+discuss the advantages of the African extension to the array in
+producing high-fidelity movies of Sgr A*.
+2. Methods
+2.1. GRMHD ground truth movies
+The quiescent state of the plasma flow of Sgr A* was reproduced
+by generating synthetic data from general relativistic magneto-
+hydrodynamic (GRMHD) simulations at 230 GHz. The typical
+range of simulations used to study Sgr A* include two classes of
+models: magnetically arrested disk (MAD; Igumenshchev et al.
+2003; Narayan et al. 2003) and Standard And Normal Evolu-
+tion (SANE; Narayan et al. 2012) models. The SANE mode is
+characterized by a weak and turbulent magnetic field crossing
+the hemisphere of the event horizon, while the MAD mode has
+high magnetic flux. The recent EHT Sgr A* results have shown
+that GRMHD simulations are more variable than the data (Event
+Horizon Telescope Collaboration et al. 2022d). Because SANE
+models are less variable than MAD models, they are more rep-
+resentative of the degree of variability in Sgr A*. We thus used a
+SANE model for our quiescent state reconstructions.
+The simulation was generated with the GRMHD code BHAC
+Porth et al. (2017); Olivares et al. (2020). We initialized a torus
+in hydrodynamic equilibrium where the inner edge is located
+at 6 M (where M is the gravitational timescale GM/c3) and the
+pressure maximum is found at 13 M. We set a black hole spin
+of a⋆ = 0.9375 and an adiabatic index ˆγ = 4/3 and per-
+formed the simulations on spherical grid (r, θ, φ) with resolution
+of 512 × 192 × 192 and three layers of adaptive mesh refinement
+(AMR) using logarithmic Kerr-Schild coordinates. For more de-
+tails on the simulations see Fromm et al. (2022). We evolve the
+simulations until 30000 M, which ensures a quasi steady-state in
+the mass accretion rate. The radiative transfer calculations were
+performed with the GRRT code BHOSS Younsi et al. (2012, 2016,
+2020, 2021). We used a field of view of 200 µas together with
+a black hole mass of 4.14 × 106 M⊙ at a distance of 8.127 kpc
+(Event Horizon Telescope Collaboration et al. 2022d). The im-
+ages were created assuming a viewing angle ϑ = 10◦ and a nu-
+merical resolution of 4002 pixels. Since the electron temperature
+is not evolved during the GRMHD simulations we computed
+their temperature using the R − β description of Mo´scibrodzka
+et al. (2016) where we set Rlow = 1 and Rhigh = 5. In order to
+adjust the simulations to the observations, we iterated over the
+mass accretion rate to provide an average flux density of ∼2.4
+Jy at 230 GHz in a time window of 5000 M. Two time windows
+were used (20-25 kM and 25-30 kM) and individually normal-
+ized.
+The 16-hour movie consists of 300 frames separated by 200
+seconds, with a rotation period of the plasma around the black
+hole of ∼30 minutes. The simulation does not include effects
+Article number, page 2 of 11
+
+Noemi La Bella et al.: Sgr A* dynamical imaging with an African extension to the EHT
+of interstellar scattering, therefore we characterized those effects
+using a phase screen toward Sgr A* (see Psaltis et al. 2018; John-
+son et al. 2018, for more details).
+2.2. Synthetic data generation
+The GRMHD synthetic data were produced with the SYMBA 1
+software (Roelofs et al. 2020), which reconstructs a model im-
+age following the same calibration and imaging processes of a
+realistic observation. Given a VLBI array configuration and a
+specific model as input, the synthetic observations are generated
+with MeqSilhouette (Blecher et al. 2017; Natarajan et al. 2022)
+and the corrupted raw data are then processed with the VLBI
+data calibration pipeline rPICARD (Janssen et al. 2019), which
+is used to calibrate real EHT data (Event Horizon Telescope Col-
+laboration et al. 2019b). The calibrated data set can also pass
+through the network calibration step that solves gains for colo-
+cated sites using the flux of the source at large scales (Fish et al.
+2011; Johnson & Gwinn 2015; Blackburn 2019; Event Horizon
+Telescope Collaboration et al. 2019c). Our synthetic data are
+based on the antenna and weather parameters as measured in
+the 2017 observations (Event Horizon Telescope Collaboration
+et al. 2019d). The weather conditions were extracted from the
+VLBI monitor server 2, which collects weather data (e.g., ground
+pressure, ground temperature) from in situ measurements. The
+weather conditions used are reported in Table 2 of Roelofs et al.
+(2020), which includes the parameters for the stations that joined
+the 2017 EHT campaign, and those for the enhanced array, with
+GLT joining the array in 2018, NOEMA and KP in 2021, and
+with the planned AMT. As described by the authors, the weather
+parameter estimation for stations that did not join the 2017 obser-
+vations was done using the Modern-Era Retrospective Analysis
+for Research and Applications, version 2 (MERRA − 2) from the
+NASA Goddard Earth Sciences Data and Information Services
+Center (Gelaro et al. 2017), and the am atmospheric model soft-
+ware Paine (2019). We applied the same method to obtain the
+weather conditions on La Palma, in the Canary Islands. Finally,
+we adopted the observing schedule of 2017 April 7 (Event Hori-
+zon Telescope Collaboration et al. 2022c), encompassing scans
+on Sgr A* from the 4 to 15 UT hours.
+For generating movies of flares in Sgr A*, we used a sim-
+ulated Gaussian flaring feature with an orbiting period of 27
+min around a ray-traced image of a semi-analytic advection-
+dominated accretion flow (ADAF) model of Sgr A* (model B
+of Doeleman et al. 2009). The movie at 230 GHz is composed
+of 100 frames separated by 16.2 seconds. The eht-imaging
+Python library 3 (Chael et al. 2016, 2018) was used to gener-
+ate the hotspot synthetic data. The eht-imaging package does
+not produce realistic VLBI-mm observations as SYMBA, for
+instance the data are not frequency-resolved, gain effects are
+not based on physical models, and there are no calibration ef-
+fects added (more details about the difference between the two
+pipelines can be found in Event Horizon Telescope Collabora-
+tion et al. 2019c, Appendix C). As in the case of the GRMHD
+movies, the synthetic data were based on the 2017 April 7 ob-
+serving parameters. Unlike SYMBA, the simulated visibilities
+are not scan-separated, but have a cadence of 30 seconds.
+1 https://bitbucket.org/M_Janssen/symba
+2 https://bitbucket.org/vlbi
+3 https://github.com/achael/eht-imaging
+2.3. Dynamical imaging
+We
+imaged
+the
+SYMBA
+synthetic
+data
+set
+using
+the
+eht-imaging library, developed specifically for the EHT. The
+imaging algorithm utilizes the regularized maximum likelihood
+(RML) method, which aims to find an image that minimizes a
+specified objective function, consisting of data fit quality (χ2)
+terms, and additional regularizer terms favoring, for example,
+smooth or sparse image structures (Event Horizon Telescope
+Collaboration et al. 2019e). The static assumption based on the
+Earth rotation aperture synthesis technique, where the source is
+assumed static during the course of the observation, is not valid
+in the case of Sgr A* due to its intraday variability (Event Hori-
+zon Telescope Collaboration et al. 2022b). To tackle this chal-
+lenge, we use a method called “dynamical imaging." The dy-
+namical imaging algorithm within the eht-imaging package is
+a generalization of the standard RML method which introduces
+three dynamical regularizers that enforce time-sensitive proper-
+ties between snapshot frames (see Johnson et al. 2017, for more
+details). To reconstruct the hotspot movies we used the R∆t reg-
+ularizer, which imposes a time continuity between frames. Since
+the hotspot model simulates coherent motion of a flare orbit-
+ing Sgr A*, this regularizer let us reconstruct continuous motion
+of structure. For the GRMHD movies, we also added the R∆I
+regularizer, which enforces similarity between the reconstructed
+frame and a time-averaged image. As GRMHD simulations de-
+scribe the turbulent behavior of an accretion flow onto Sgr A*,
+this regularizer allows us to look for turbulence on top of a static
+structure.
+To inspect the capability of the expanded EHT array to re-
+construct dynamical motion, we selected time windows during
+the observation for which coverage and filling fraction were op-
+timized, as was done in Farah et al. (2022). For the GRMHD sim-
+ulations, we produced movies of 5.7 hours, while for the hotspot
+movies we chose optimal time windows of 1.7 hours where the
+array offers the best coverage. To obtain a good reconstructed
+movie, larger time windows were required for the GRMHD data
+set generated with SYMBA, which includes actual scans and
+gaps between the scans (more details in Section 3.1).
+2.4. Movie quality metrics
+Two quality metrics were selected to evaluate the fidelity of
+the reconstructed images: the normalized root-mean-square er-
+ror (NRMSE) and the normalized cross-correlation (NXCORR;
+e.g., Event Horizon Telescope Collaboration et al. 2019e).
+NRMSE is more sensitive to pixel-by-pixel differences, while
+NXCORR is more sensitive to large scale structure (Issaoun et al.
+2019b). We estimated values for both metrics for each frame
+of the movie, quantifying the fidelity of the reconstruction as
+a function of time with respect to the ground truth.
+The NRMSE measures similarities per kth pixel and it is de-
+fined as:
+NRMSE =
+��
+k(Ik − I′
+k)2
+�
+k I2
+k
+,
+(1)
+where I′ and I are the intensity of the reconstructed movie frame
+and the model movie frame, respectively (e.g., Chael et al. 2018;
+Issaoun et al. 2019b). An NRMSE value of zero corresponds to
+identical images.
+For given frames I′ and I, NXCORR is given by:
+NXCORR = 1
+N
+�
+k
+(Ik − ⟨I⟩)(I′
+k − ⟨I′⟩)
+σIσI′
+,
+(2)
+Article number, page 3 of 11
+
+A&A proofs: manuscript no. core
+Fig. 1: Sgr A* (u, v) coverage of the 2017 April 7 EHT observa-
+tions. Seven scans on Sgr A* were added to the original schedule
+at the beginning of the observation, brought by the introduction
+of the NOEMA array and the African stations. In blue, the cov-
+erage obtained with the 2022EHT array. The contributions of the
+AMT and CNI baselines are shown in red and in brown, respec-
+tively. The AMT adds long northeast and southwest baselines in-
+creasing the EHT resolution, while CNI offers shorter baselines
+to detect large-scale emission and constrain the source extent.
+where N is the total number of pixels per frame, ⟨I⟩ and ⟨I′⟩ are
+the mean pixel values and σI, σI′ are the respective standard de-
+viations. An NXCORR of 1 corresponds to a perfect correlation
+between the frames, -1 for anticorrelation, and 0 for no correla-
+tion (e.g., Event Horizon Telescope Collaboration et al. 2019e).
+3. The African expansion to the EHT
+In this section, we discuss a potential implementation of the
+African expansion to the EHT, its (u, v) coverage, and Fourier
+filling fraction, which let us identify potential time windows to
+generate movies of Sgr A*. We also investigate the location of
+the new baselines with respect to the position of the two local
+minima in the correlated flux density profile of a thin ring of 54
+µas. To assess the impact of the new African stations, different
+array configurations were used. We name those configurations as
+follows: 2022EHT, the current EHT configuration composed of
+eleven telescopes; 2022EHT + AMT, the 2022EHT with the ad-
+dition of AMT; 2022EHT + Africa, the 2022EHT plus the AMT
+and CNI stations; Eastern array + Africa, the 2022EHT subar-
+ray until ∼9.5 UT hours (∼22.7 Greenwich Mean Sidereal Time,
+GMST), after this time the AMT does not observe Sgr A*; West-
+ern array, the 2022EHT subarray from ∼9.5 UT hours to ∼15 UT
+hours (∼4.1 GMST). So far, the Western array has been offering
+the best coverage to produce dynamic reconstructions of Sgr A*.
+3.1. (u, v) coverage
+Fig. 1 depicts the Sgr A* (u, v) coverage using the 2017 April
+7 observing schedule as a base, enhanced by the addition of
+NOEMA and KP, which joined the array post-2017, and the two
+proposed African antennas. Moreover, the observation was im-
+posed to start when the source is at an elevation of more than
+10 degrees at NOEMA and the African telescopes, allowing us
+to extend the observation by two hours. The 2022EHT baselines
+are shown in blue, the AMT baselines in red and the CNI base-
+lines in brown. The AMT is a potential southern site to image
+Sgr A* that adds determinant baselines to the array. Specifically,
+the AMT adds northsouth baselines to PV and NOEMA, east-
+west baselines to Chile, and a redundancy baseline to ALMA-
+SPT, since Mt. Gamsberg is at the same latitude as ALMA.
+Moreover, the AMT increases the resolution in the northeast and
+southwest, by adding long baselines to LMT and the Arizona
+stations. On the other hand, the CNI telescope yields new short
+inter-site baselines to the European sites, PV and NOEMA, con-
+tributing to the measurement of the source extent, together with
+the inter-sites SMT-LMT, PV-NOEMA baselines. In addition,
+the baseline CNI-AMT provides further northsouth coverage to
+the array.
+3.2. Fourier filling fraction
+The sparsity and changing coverage of the EHT array affect
+the accuracy of the dynamical reconstructions of time-variable
+sources. To produce VLBI movies of Sgr A*, it is thus required
+to identify time periods with optimal and stable (u, v) coverage.
+For the 2017 Sgr A* results, Farah et al. (2022) selected time re-
+gions using three different metrics and found the best dynamical
+time period to be from ∼01:30 GMST to ∼03:10 GMST, hence in
+the Western array window. We utilized one of these metrics, the
+(u, v) filling fraction, to inspect if new temporal regions are of-
+fered by the Eastern array + Africa. The Fourier filling fraction
+measures the area sampled in the (u, v) plane by the observed
+visibilities. Following Farah et al. (2022), the (u, v) points were
+convolved with a circle of radius 0.71/θFOV, with FOV being the
+field of view adopted for imaging, representing the half-width at
+half-maximum of a filled disk of uniform brightness on the sky
+(see Palumbo et al. 2019, for more details). In our analysis, we
+calculated the filling fraction normalized to the maximum fill-
+ing fraction value of the 2022EHT array. On the left of Fig. 2,
+we show the time-dependent normalized filling fraction for the
+2022EHT + AMT array in red, and that of the 2022EHT array in
+blue. The colored windows delimit time regions in which the fill-
+ing fraction is persistently above the 70% 2022EHT maximum
+threshold (dashed grey line). Time windows below this threshold
+do not have sufficient coverage to produce high-fidelity movies.
+The 2022EHT array provides good time regions in the Western
+array. Notably, our results confirm the 01:30 GMST to 03:10
+GMST best-time window obtained from the 2017 array selective
+dynamical imaging analysis (Farah et al. 2022). The AMT adds
+three additional optimal time periods (red areas) in the Eastern
+array, of almost 4 hours in total. Furthermore, on the right of Fig.
+2 we show a further increase in the Fourier filling area achieved
+by the combination of the CNI (brown) and AMT sites (i.e., with
+the 2022EHT array + Africa) leading to a persistent time block
+of 7.4 hours. Therefore, the African stations will provide signif-
+icantly improved (u, v) coverage and stability for the Eastern ar-
+ray, increasing the ability to study rapid variations of the source
+at the beginning of the observing track.
+Article number, page 4 of 11
+
+10
+CNI baselines
+AMT baselines
+7.5
+2022EHT
+5.0
+2.5
+[G^]
+0.0
+-2.5
+-5.0
+-7.5
+-10
+-10
+-7.5
+-5.0
+-2.5
+0.0
+2.5
+5.0
+7.5
+10
+u [G入]Noemi La Bella et al.: Sgr A* dynamical imaging with an African extension to the EHT
+Fig. 2: Time-dependent Fourier filling fraction normalized by the maximum Fourier filling of the 2022EHT array. The curves
+represent the filling fraction of the 2022EHT array, 2022EHT + AMT array and 2022EHT + Africa array, in blue, red and brown,
+respectively. The dashed gray line corresponds to the lower limit used for identifying good time windows to perform dynamical
+imaging. The optimal time regions for the current EHT array are shown in blue. The 2022EHT + AMT adds three time windows
+(red areas) of ∼4 hours in total, while the 2022EHT + Africa array (brown area) produces a time window of ∼7.4 hours.
+3.3. Correlated flux density profile
+The correlated flux density (in Jy) of Sgr A* as a function of pro-
+jected baseline length was investigated for both the Eastern and
+Western arrays using the network calibrated data sets obtained as
+output of SYMBA. The calibrated amplitudes of April 7, shown
+in Fig.3a for the Eastern array and in Fig.3b for the Western ar-
+ray, resemble a Bessel function with a first null at ∼3.0 Gλ and
+a second null at ∼6.5 Gλ, corresponding to a thin ring with a
+54 µas diameter (Event Horizon Telescope Collaboration et al.
+2022b). In Fig.3a, the African baselines, which are represented
+in orange, probe the prominent secondary peak. The African
+stations also provide short inter-site baselines at the same pro-
+jected baseline length as the SMT-LMT baseline, highlighted in
+cyan in Fig.3b. In 2017, the SMT-LMT baseline was the short-
+est inter-site baseline in the EHT array, which yields the size
+and the compact flux density estimation of the source (Issaoun
+et al. 2019b). However, 2017 EHT observations have shown that
+LMT is a challenging station to calibrate and the determination
+of the compact flux is required to establish constraints on the data
+(Event Horizon Telescope Collaboration et al. 2019e, 2022b).
+Since 2021, NOEMA and KP have added short baselines to PV
+and SMT, respectively, useful for amplitude calibration. Thus,
+the African baselines shorter than 2Gλ are important for the EHT
+imaging process as they can contribute to compute the size and
+the total compact flux density of the source.
+4. Results from imaging
+From the filling fraction study with the 2022EHT array + Africa,
+we estimated new time regions offered in the Eastern array to
+perform dynamical imaging. Here, we show the static and dy-
+namical reconstructions from the GRMHD datasets generated
+with SYMBA using the Eastern array + Africa and Western ar-
+ray. Moreover, we present the dynamical reconstructions ob-
+tained from the hotspot model, which lets us test the capability
+of the array to image coherent motion or flares in Sgr A*. Unlike
+the (u, v) coverage inspection, the following images are obtained
+without the additional 2 hours observing Sgr A* provided by the
+African stations. In this way, we compare the capabilities of the
+two subarrays to image Sgr A* for the same observing time.
+4.1. GRMHD static reconstructions
+Fig. 4 shows the static images reconstructed from the GRMHD
+datasets for the different array configurations. The synthetic im-
+ages were compared with the time-averaged image of the SANE
+simulation (first column), which was convolved with a Gaus-
+sian kernel with Full Width Half Maximum (FWHM) of 0.6 ×
+clean beam. As described in Sec. 2.3, the static images were pro-
+duced using the eht-imaging package. We corrected for the ef-
+fect of the diffractive scattering with the eht − imaging deblur
+function (Event Horizon Telescope Collaboration et al. 2022b),
+which divides the interferometric visibilities by the Sgr A* scat-
+tering kernel.
+Because the Eastern array without the African stations does
+not have sufficient coverage toward Sgr A*, as we note from the
+filling fraction analysis, it is not able to resolve its black hole
+shadow. The static reconstruction of Sgr A* significantly im-
+proves when the AMT is added to the Eastern array, producing
+an image with a clear evidence of the ring-like structure. The im-
+age robustness increases with the Eastern array + Africa, indeed
+the artifacts present in the northwest and northeast of the ring
+are less evident than in the Eastern array + AMT image. The av-
+eraged reconstruction using the Western array is also illustrated
+in the right-most side of the figure. The subarray is capable of
+reconstructing the black hole shadow, but with a lower quality
+than the Eastern array with the African stations. The fidelity of
+Article number, page 5 of 11
+
+1.4
+2022EHT+AMT
+2022EHT
+1.2
+Normalized filling fraction
+1.0
+0.8
+0.6
+0.4
+2.4 hr
+1.0hr 0.8 hr
+1.0hr
+2.0 hr
+0.2
+17
+19
+21
+23
+1
+3
+Time GMST (hr)2022EHT+Africa
+2022EHT+AMT
+2022EHT
+1.2
+1.0
+0.8
+0.6
+0.4
+7.4 hr
+1.0hr
+2.0 hr
+0.2
+17
+19
+21
+23
+1
+3
+Time GMST (hr)A&A proofs: manuscript no. core
+0
+2
+4
+6
+8
+Projected Baseline Length (G )
+10
+4
+10
+3
+10
+2
+10
+1
+100
+Correlated Flux Density (Jy)
+Africa baselines
+other baselines
+(a)
+0
+2
+4
+6
+8
+Projected Baseline Length (G )
+10
+4
+10
+3
+10
+2
+10
+1
+100
+Correlated Flux Density (Jy)
+SMT-LMT baseline
+other baselines
+(b)
+Fig. 3: Correlated flux density as a function of baseline length
+for the Eastern (a) and Western (b) arrays. The African baselines
+(in orange) will contribute to probe the secondary peak, but also
+add short baselines to the array, at comparable projected baseline
+lengths to the SMT-LMT baseline (cyan). The shortest inter-site
+baselines are needed to estimate the extent and the total compact
+flux density of the source.
+the reconstructions using the different array configurations well
+represents the filling fraction trend reported in Fig. 2 and dis-
+cussed in Sec. 3.2.
+In typical static imaging, the full observing track is used
+to produce the final averaged image. In Fig. 4, we show seg-
+mented time-averaged reconstructions obtained with the East-
+ern and Western arrays individually with the purpose of examin-
+ing the African station impact on imaging the static structure of
+Sgr A*. The high-fidelity average image from the full 2022EHT
++ Africa array is illustrated on the left of Fig. 5, while on the
+right we show the static reconstruction using the 2022EHT ar-
+ray (see for comparison the representative model of Sgr A* in
+Fig. 4). The 2022EHT + Africa average image is used as the
+prior and initial image for the RML dynamical imaging of the
+GRMHD data sets presented in the next section.
+4.2. GRMHD dynamical reconstructions
+Movies of Sgr A* were produced with the dynamical imaging
+algorithm introduced in Sec. 2. Based on the candidate time re-
+gions with good (u, v) coverage explored in Sec. 2, we produced
+movies for the Eastern and Western arrays, separately. To per-
+form dynamical imaging on the GRMHD data sets, which con-
+tain visibilities on a scan basis, we chose large time periods of
+∼5.7 hours, specifically from 17 GMST to 22.7 GMST for the
+Eastern array and from 22.7 GMST to 4.1 GMST for the West-
+ern array. The visibilities were averaged every 1 min to enhance
+the signal-to-noise ratio. The GRMHD simulation movie, which
+has a frame duration of 200 seconds, was synchronized to the
+reconstructed movies, which have a frame separation of 1 min.
+The synchronized model movie was created by averaging over
+the model frames that fall between the start and the end of the
+observed frame. In this way, we could estimate the NRMSE and
+NXCORR between the ground truth movie and the reconstructed
+movie frame by frame and select the data term and regularizer
+weights that minimize the NRMSE and maximize the NXCORR.
+In Fig. 6, we illustrate five snapshots of the movies recon-
+structed for the Eastern array + Africa (second row) and for the
+Western array (fourth row), and the corresponding frames of the
+SANE model. Each snapshot timestamp is shown at the top of
+the images. As for the static imaging, the reconstructions are
+descattered, by deblurring the interferometric data. The model
+movie was blurred using a Gaussian with a FWHM of 0.9 ×
+clean beam of the 2022EHT + Africa data sets, while the recon-
+structions were blurred with a FWMH of 0.6 × clean beam. A
+lower blurring fraction is needed for the reconstructions because
+the dynamical imaging process inherently produces smoother
+structure.
+The dynamical reconstructions generated with the Eastern
+array + Africa reproduce accurately the ring-like structure of the
+GRMHD simulation, while a less solid performance is obtained
+with the Western array. The reported NXCORR values in the
+bottom of the images confirm the robustness of the Eastern array
++ Africa reconstructions. The NRMSE values are also consistent
+with the general goodness trend of the reconstructions.
+We use GRMHD simulations of Sgr A* to test if the East-
+ern array + Africa is able to reconstruct the main ring structure
+and its brightness distribution. GRMHD models reproduce a qui-
+escent yet turbulent accretion flow and are not representative of
+coherent motion of features expected in the event of flaring activ-
+ity. Moreover, GRMHD models are complex and challenging to
+reconstruct due to the large amplitudes in the variability (Event
+Horizon Telescope Collaboration et al. 2022d), making it diffi-
+cult to recognize the rotation of individual features. Dynamical
+imaging using a simple hotspot model, shown in the next sec-
+tion, allows us to easily investigate the capability of the array to
+reconstruct coherent motion in Sgr A* in the event of flares.
+4.3. Hotspot dynamical reconstructions
+Fig. 7 shows five snapshots of the dynamical reconstructions
+generated using as ground truth the hotspot crescent model.
+In the first row, we present the synchronized model snapshots,
+while the Eastern array + Africa and Western array dynamical
+reconstructions are illustrated in the second and third row, re-
+spectively. Similarly to the GRMHD models, we identified the
+data terms and regularizer weights that maximize the similari-
+ties between the model and the reconstruction snapshots, exploit-
+ing the NXCORR and NRMSE metrics. Unlike the GRMHD re-
+constructions, the visibilities are separated by ∼30 seconds and
+the dynamical imaging was performed in narrow time regions of
+about 1.7 hours. In particular, for the Eastern array + Africa this
+was chosen to be from 21 to 22.7 GMST, which corresponds to
+the best time window offered by the subarray. For the Western
+array, the best period is given between the 1.5 and 3.2 GMST.
+The five snapshots are separated by almost 0.1 hour in order to
+represent the hotspot orbit, which is completed in ∼0.5 hours
+(i.e., 27 minutes). As confirmed by the NXCORR (reported in
+the figure) and the NRMSE, the individual frames produced in
+Article number, page 6 of 11
+
+Noemi La Bella et al.: Sgr A* dynamical imaging with an African extension to the EHT
+Fig. 4: Time-averaged reconstructions of Sgr A* obtained from the GRMHD synthetic observations for the different array configura-
+tions. The leftmost image shows the static representation of the GRMHD simulation used as ground truth movie. The Eastern array
+without the AMT (second image) does not resolve the shadow of the black hole. The addition of the AMT significantly impacts
+the fidelity of the reconstruction, and a further improvement is obtained with the African array (third image). The rightmost image
+shows the averaged reconstruction produced using the Western array alone. The images were blurred with a Gaussian FWHM equal
+to 0.6 × clean beam of the 2022EHT + Africa data set.
+Fig. 5: Time-averaged reconstructions of GRMHD simulations
+of Sgr A* with the 2022EHT + Africa and 2022EHT arrays.
+The ground truth model is shown in the first column of Fig. 4.
+The 2022EHT + Africa array produces a higher fidelity image,
+which is used as the prior for the dynamical imaging.
+the Eastern array + Africa time region are more accurate than
+in the Western array window. Indeed, in the latter, the snapshots
+present pronounced northeast and southwest imaging artifacts.
+Both subarrays are capable of reconstructing the motion of the
+hotspot, confirming that the addition of the African stations to
+the EHT array provides a new time window in the first half of the
+observation to detect rapid coherent flux variations in Sgr A*’s
+accretion flow or jet. In order to effectively establish the capa-
+bility of the array in reproducing the flare motion, we developed
+two methods that evaluate the robustness of our dynamical im-
+ages. For the two subarrays, we investigate the ability to recover
+the flux density profile and the time-dependent rotational veloc-
+ity of the hotspot. The two methods are described in Sec. 4.3.1
+and in Sec. 4.3.2.
+4.3.1. Method 1: Flux density profile
+To assess the ability of the Eastern array + Africa to reconstruct
+the flux density around the crescent model, we calculated the
+flux density pixel by pixel as a function of the position angle
+for each snapshot. We selected the ring from which to extract
+the flux using the hough_ring function in the eht-imaging
+library, which finds circles in an image according to the pixel
+brightness distribution. The choice was made giving as input the
+time-averaged model image. Thus, for each model snapshots and
+reconstructed frames, the flux density was estimated within a ra-
+dius of 32µas and in sectors 10 degrees wide. In Fig. 8, we show
+the flux density as a function of the angle for five snapshots of
+the ground truth model (in green and also illustrated in the lower
+panel of the image), of the Eastern array + Africa movie (in red)
+and of the Western array movie (in blue). Because of the asym-
+metry of the brightness distribution in the crescent model, the
+flux profile has a peak in the snapshots when the hotspot is at its
+maximum intensity (i.e., third column), while it decreases when
+the hotspot is located on the opposite side. From the model snap-
+shots and the corresponding flux density profile, we note that the
+angular position of the hotspot is correctly determined by this
+method. The flux density profiles obtained with the Eastern ar-
+ray + Africa and Western array recover quite well the hotspot
+motion, both in term of intensity and in position angle.
+4.3.2. Method 2: Rotational velocity profile
+Additionally, we computed the rotational velocity of the hotspot
+as a function of time. This rotation (in degrees per minute) is de-
+fined as the degree of rotation for each frame i with respect to the
+fifth subsequent frame j. In order to measure it, we rotated frame
+i in steps of two degrees across a range of angles. We calculated
+the NXCORR (i.e., the image correspondence) with respect to
+frame j at each rotation angle. The angle at which the NXCORR
+is maximized between the two frames divided by the time dura-
+tion between frames i and j gives us the rotational velocity. We
+measured the rotational velocity of the hotspot every five frames,
+which lets us reconstruct its motion. As the hotspot completes its
+orbit every 27 min and the frame separation of the reconstructed
+movie is ∼30 seconds, the rotation every five frames (∼33◦) is
+easier to measure than the rotation per frame (∼6.6◦).
+The rotational velocity obtained for the Eastern array +
+Africa and the Western array movies are shown in the left and
+right of Figure 9 in red and in blue, respectively. The hotspot ve-
+locity measured from the model movie is represented in green.
+As in the case of the flux profile, the method represents the asym-
+metric brightness distribution of the crescent model. Indeed, the
+frames with the maximum intensity of the hotspot have a maxi-
+mum value of the rotational velocity, which drops to zero when
+the hotspot is not present. The negative values of the velocity
+Article number, page 7 of 11
+
+Model
+Eastern
+Eastern + AMT
+Eastern + Africa
+Western
+0
+0
+60 μas
+0.5
+1.0
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+1.5
+Tb[K]
+1e10
+Tb[K]
+1e10
+Tb[K]
+1e10
+Tb[K]
+1e10
+Tb[K]
+1e102022EHT + Africa
+2022EHT
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+T,[K]
+1e10
+T,[K]
+1e10A&A proofs: manuscript no. core
+Fig. 6: Dynamical reconstructions obtained from the GRMHD data sets. The first row shows five snapshots of the GRMHD sim-
+ulation taken in the Eastern array (17-22.7 GMST), the second row represents the respective dynamical reconstructions using the
+Eastern array + Africa. In a similar way, the third row and forth row illustrate the GRMHD frame simulations and the correspondent
+frame reconstructions using the Western array (22.7-4.1 GMST). The blurring utilized for the GRMHD simulation is 0.6 × clean
+beam. Higher quality dynamical reconstructions are produced by the Eastern array + Africa, also confirmed by the NXCORR metric
+reported at the bottom of each image. The numbers on the top of the GRMHD simulation snapshots represent the frame time.
+are artifact produced by the method. In particular, these unphys-
+ical features are generated for each period of the hotspot movie,
+when we compare the last frame that contains the hotspot and the
+fifth frame that presents only the crescent emission. Comparing
+the rotational velocity curves derived from the Eastern array +
+Africa and the Western array movies with the model simulation,
+we find that the flare variability is most accurately recovered in
+the Eastern time window.
+5. Summary and conclusions
+We generated synthetic data of Sgr A* with the current EHT
+array and two stations in the African continent, the AMT and
+the CNI telescope. We have evaluated the capability of the
+EHT Eastern subarray with the African sites (17-22.7 GMST)
+to produce movies of Sgr A* and compared it to the Western
+subarray (22.7-4.1 GMST). The data sets were created from
+ray-traced images of a SANE GRMHD simulation, which is
+representative of the quiescent yet turbulent black hole accretion
+Article number, page 8 of 11
+
+GRMHD simulation
+18.0 GMST
+19.1 GMST
+21.3 GMST
+21.8GMST
+22.3GMST
+60 μas
+Eastern array + Africa
+NXCORR0.994
+NXCORR0.993
+NXCORR0.985
+NXCORR0.990
+NXCORR0.993
+GRMHD simulation
+23.2 GMST
+0.4 GMST
+1.4 GMST
+1.8GMST
+3.1 GMST
+C
+Westernarray
+NXCORR0.981
+NXCORR0.987
+NXCORR0.990
+NXCORR0.986
+NXCORR0.974Noemi La Bella et al.: Sgr A* dynamical imaging with an African extension to the EHT
+Fig. 7: Dynamical reconstructions generated using the hotspot synthetic data. In the first row we show five snapshots of the hotspot
+model movie. The hotspot performs a full rotation every 27 mins. The frames were chosen to represent a complete orbit. The
+reconstructions obtained from the dynamical imaging using the Eastern array + Africa and Western array are shown in the second
+and third row, respectively. The movies were generated in a time window of about 1.7 hours (21-22.7 GMST for the Eastern array,
+1.5-3.2 GMST for the Western). The NXCORR values estimated for the reconstructions is reported in the bottom of each images.
+The temporal evolution is available as an online movie.
+flow, and from a crescent hotspot model to test the imaging
+performance of the array in reconstructing coherent motion
+from flaring activity in Sgr A*.
+We found that the AMT increases the resolution of the EHT
+array via long baselines with the Arizona and Mexico sites, while
+short baselines provided by the African extension to the EHT
+constrain the compactness and extent of the source on larger
+scales. We estimated the Fourier filling fraction with the EHT ar-
+ray and the Africa telescopes to investigate the presence of good
+time regions to perform dynamical imaging. We found that the
+added baselines offer an optimal time window of about 7 hours
+in the Eastern array, allowing to produce high-fidelity movies of
+Sgr A* from the very start of a typical observing track. This in-
+creases the time in which dynamical imaging is possible by a
+factor > 4. In comparison, Farah et al. (2022) demonstrated that
+with the 2017 EHT array, the only time period in which we are
+able to reconstruct the variability of the source is from ∼01:30
+GMST to ∼03:10 GMST, with the Western array.
+Our static reconstructions of the GRMHD simulation con-
+firm the importance of the AMT in imaging Sgr A*. Without
+the AMT, the data set generated with the current EHT configu-
+ration is not able to reproduce a physical image of the black hole
+shadow in the Eastern array window. Including the African sites,
+we can perform high-fidelity imaging of Sgr A* with reduced
+artifacts. Additionally, we produced GRMHD dynamical recon-
+structions limited to the best Eastern and Western time regions.
+The African stations enable accurate frame reconstructions of
+the ring-like structure when included in the Eastern array. Since
+the rotation of individual features is difficult be recognized in the
+turbulent flow of GRMHD simulations, we performed a hotspot
+dynamical imaging analysis to test the capability of the different
+arrays to reconstruct coherent motion mimicking flaring activity
+in Sgr A*. Compared to the 2022EHT array, the African stations
+open a new time window in the Eastern array that can be used
+to reconstruct motion in the accretion disk. We developed two
+methods involving the determination of the flux density profile
+and the rotational velocity of the hotspot to establish the suc-
+cessful performance of the enhanced Eastern array in reproduc-
+ing the motion in Sgr A*. Our results show the impact of adding
+stations in the African continent in increasing the time-variable
+(u, v) coverage of the EHT toward Sgr A*. The African exten-
+sion will be crucial for future EHT observations to study accu-
+Article number, page 9 of 11
+
+Hotspot movie
+t=0.05hr
+t=0.13hr
+t=0.23hr
+t=0.31 hr
+t=0.54 hr
+70μas
+Easternarray+Africa
+NXCORR0.957
+NXCORR0.968
+NXCORR0.955
+NXCORR0.970
+NXCORR0.953
+Western array
+NXCORR0.906
+NXCORR0.923
+NXCORR0.892
+NXCORR0.906
+NXCORR0.952A&A proofs: manuscript no. core
+Fig. 8: Flux density (Jy/pixel) in function of the angle (degrees) estimated in five snapshots of the model movie (in green), of the
+Eastern array + Africa movie (in red), and of the Western array movie (in blue). The brightness distribution was estimated using a
+ring with outer radius of 32 µas, divided in sectors 10 degrees wide. The five frames of the model simulation from where the flux
+densities were extracted are shown in the bottom panel.
+Fig. 9: Rotational velocity (degree per minute) in function of the time for the Eastern + Africa array movie (left) and for the Western
+array movie (right). In green, the rotational velocity for the hotspot movie simulation. The profile were obtained by searching for the
+angle that maximize the similarity between each frame and the subsequent fifth frame. The Eastern array + Africa movie presents a
+more robust reconstruction of the hotspot rotation than the Western array. The negative values of the rotation are artifacts produced
+by the method utilized.
+rately the time-variable source at our Galactic Center through
+high-fidelity movies across an observing track.
+Acknowledgements. We thank Oliver Porth for performing the ray-tracing for
+the GRMHD simulation used. This publication is part of the project Dutch Black
+Hole Consortium (with project number 1292.19.202) of the research programme
+NWA which is (partly) financed by the Dutch Research Council (NWO). SI
+is supported by Hubble Fellowship grant HST-HF2-51482.001-A awarded by
+the Space Telescope Science Institute, which is operated by the Association of
+Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-
+26555. FR is supported by NSF grants AST-1935980 and AST-2034306, and the
+Black Hole Initiative at Harvard University, made possible through the support
+of grants from the Gordon and Betty Moore Foundation and the John Templeton
+Foundation. The opinions expressed in this publication are those of the author(s)
+and do not necessarily reflect the views of the Moore or Templeton Foundations.
+CMF is supported by the DFG research grant “Jet physics on horizon scales and
+beyond" (Grant No. FR 4069/2-1) The simulations were performed on LOEWE
+at the CSC-Frankfurt, Iboga at ITP Frankfurt and Pi2.0 at Shanghai Jiao Tong
+University.
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+On the Weihrauch degree of the additive Ramsey
+theorem
+Arno Pauly � �
+School of Mathematics and Computer Science, Swansea University, UK
+Pierre Pradic �
+School of Mathematics and Computer Science, Swansea University, UK
+Giovanni Soldà � �
+School of Mathematics and Computer Science, Swansea University, UK 1
+Abstract
+We characterize the strength, in terms of Weihrauch degrees, of certain problems related to Ramsey-
+like theorems concerning colourings of the rationals and of the natural numbers. The theorems we
+are chiefly interested in assert the existence of almost-homogeneous sets for colourings of pairs of
+rationals respectively natural numbers satisfying properties determined by some additional algebraic
+structure on the set of colours.
+In the context of reverse mathematics, most of the principles we study are equivalent to Σ0
+2-
+induction over RCA0. The associated problems in the Weihrauch lattice are related to TC∗
+N, (LPO′)∗
+or their product, depending on their precise formalizations.
+2012 ACM Subject Classification Theory of computation → Proof theory; Theory of computation
+→ Computability
+Keywords and phrases Weihrauch reducibility, Reverse mathematics, additive Ramsey, Σ0
+2-induction.
+Related Version This article extends the conference paper [15] by the second and third author.
+Funding Giovanni Soldà: This author was supported by an LMS Early Career Fellowship.
+1
+Introduction
+The infinite Ramsey theorem says that for any colouring c of n-uples of a given arity of an
+infinite set X, there exists a infinite subset H ⊆ X such that the set of n-tuples [H]n of
+elements of H is homogeneous. This statement is non-constructive: even if the colouring c is
+given by a computable function, it is not the case that we can find a computable homogeneous
+subset of X. Various attempts have been made to quantify how non-computable this problem
+and some of its natural restrictions are. This is in turn linked to the axiomatic strength
+of the corresponding theorems, as investigated in reverse mathematics [17] where Ramsey’s
+theorem is a privileged object of study [9].
+This paper is devoted to a variant of Ramsey’s theorem with the following restrictions: we
+colour pairs of rational numbers and we require some additional structure on the colouring,
+namely that it is additive. A similar statement first appeared in [16, Theorem 1.3] to give a
+self-contained proof of decidability of the monadic second-order logic of (Q, <). We will also
+analyse a simpler statement we call the shuffle principle, a related tool appearing in more
+modern decidability proofs [5, Lemma 16]. The shuffle principle states that every Q-indexed
+word (with letters in a finite alphabet) contains a convex subword in which every letter
+appears densely or not at all. Much like the additive restriction of the Ramsey theorem for
+pairs over N, studied from the point of view of reverse mathematics in [11], we obtain a neat
+correspondence with Σ0
+2-induction (Σ0
+2-IND).
+1 Soldà has since moved to Ghent University
+arXiv:2301.02833v1  [cs.LO]  7 Jan 2023
+
+2
+On the Weihrauch degree of the additive Ramsey theorem
+▶ Theorem 1. In the weak second-order arithmetic RCA0, Σ0
+2-IND is equivalent to both the
+shuffle principle and the additive Ramsey theorem for Q.
+We take this analysis one step further in the framework of Weihrauch reducibility that al-
+lows to measure the uniform strength of general multi-valued functions (also called problems)
+over Baire space. Let Shuffle and ARTQ be the most obvious problems corresponding to the
+shuffle principle and additive Ramsey theorem over Q respectively. We relate them, as well
+as various weakenings cShuffle, cARTQ, iShuffle and iARTQ that only output sets of colours
+or intervals, to the standard (incomparable) problems TCN and LPO′. We also consider the
+ordered Ramsey principle, ORTQ, where the colours k come equipped with a partial order
+⪯, and the colouring α : [Q]2 → k satisfying that α(r1, r2) ⪯ α(q1, q2) if q1 ≤ r1 < r2 ≤ q2.
+A weakening of Shuffle is the principle (η)1
+<∞ introduced in [8] where we ask merely for an
+interval where some colour is dense; respectively for a colour which is dense somewhere.
+▶ Theorem 2. We have the following equivalences
+Shuffle ≡W ARTQ ≡W TC∗
+N × (LPO′)∗
+cShuffle ≡W cARTQ ≡W (LPO′)∗
+iShuffle ≡W iARTQ ≡W (η)1
+<∞ ≡W i(η)1
+<∞ ≡W TC∗
+N
+ORTQ ≡W LPO∗
+c(η)1
+<∞ ≡W cRT1
++
+Finally, we turn to carrying out the analysis of those Ramseyan theorems over N in
+the framework of Weihrauch reducibility. The additive Ramsey theorem over N is also an
+important tool in the study of monadic second order logic over countable scattered orders. As
+for the case of Q, we relate problems ARTN and ORTN as well as some natural weakenings
+cARTN, cORTN, iARTN and iORTN, to TCN and LPO′ (the i variants of those principle
+return, rather than an interval, some upper bound n on the first two points of some infinite
+homogeneous set).
+▶ Theorem 3. We have the following equivalences
+ORTN ≡W ARTN ≡W TC∗
+N × (LPO′)∗
+cORTN ≡W cARTN ≡W (LPO′)∗
+iORTN ≡W iARTN ≡W TC∗
+N
+2
+Background
+In this section, we will introduce the necessary background for the rest of the paper, and
+fix most of the notation that we will use, except for formal definitions related to weak
+subsystems of second-order arithmetic, in particular RCA0 (which consists of Σ0
+1-induction
+and recursive comprehension) and RCA0 + Σ0
+2-IND. A standard reference for that material
+and, more generally, systems of interest in reverse mathematics, is [17].
+2.1
+Generic notations
+We identify k ∈ N with the finite set {0, . . . , k − 1}. For every linear order (X, <X), we
+write [X]2 for the set of pairs (x, y) with x <X y. In this paper, by an interval I we always
+mean a pair (u, v) ∈ [Q]2, regarded as the set ]u, v[ of rationals; we never use interval with
+irrational extrema. Finally, for any sequence of elements (Xn)n∈N of elements taken from a
+poset, write lim sup(X) for infk∈N supn≥k Xn.
+
+A. Pauly, P. Pradic & G. Soldà
+3
+2.2
+Additive and ordered colourings
+For the following definition, fix a linear order (X, <X). For every poset (P, ≺P ), we call a
+colouring c : [X]2 → P ordered if we have c(x, y) ⪯P c(x′, y′) when x′ ≤X x <X y ≤X y′.
+Call c right-ordered if we have c(x, y) ⪯P c(x, y′) when x <X y ≤X y′ (in particular being
+right-ordered is less restrictive than being ordered). A colouring c : [X]2 → S is called
+additive with respect to a semigroup structure (S, ·) if we have c(x, z) = c(x, y) · c(y, z)
+whenever x <X y <X z. A subset A ⊆ X is dense in X if for every x, y ∈ A with x <X y
+there is z ∈ A such that x <X z <X y. Given a colouring c : [X]n → k and some interval
+Y ⊆ X, we say that Y is c-densely homogeneous if there exists a finite partition of Y into
+dense subsets Di such that each [Di]n is monochromatic (that is, |c([Di]n)| ≤ 1). We will
+call those c-shuffles if c happens to be a colouring of Q (i.e. X = Q and n = 1). Finally,
+given a colouring c : Q → k, and given an interval I ⊆ Q, we say that a colour i < k occurs
+densely in I if the set of x ∈ Q such that c(x) = i is dense in I.
+▶ Definition 4. The following are statements of second-order arithmetic:
+ORTQ: for every finite poset (P, ≺P ) and ordered colouring c : [Q]2 → P, there exists a
+c-homogeneous interval ]u, v[ ⊂ Q.
+Shuffle: for every k ∈ N and colouring c : Q → k, there exists an interval I = ]x, y[ such
+that I is a c-shuffle.
+ARTQ: for every finite semigroup (S, ·) and additive colouring c : [Q]2 → S, there exists
+an interval I = ]x, y[ such that I is c-densely homogeneous.
+ORTN: for every finite poset (P, ≺P ) and right-ordered colouring c : [Q]2 → P, there
+exists an infinite c-homogeneous set.
+ARTN: for every finite semigroup (S, ·) and additive colouring c : [N]2 → S, there is an
+infinite c-homogeneous set.
+As mentioned before, a result similar to ARTQ was originally proved by Shelah in [16,
+Theorem 1.3 & Conclusion 1.4] and Shuffle is a central lemma when analysing labellings of
+Q (see e.g. [5]). We will establish that ARTQ and Shuffle are equivalent to Σ0
+2-induction over
+RCA0 while ORTQ is provable in RCA0.
+We introduce some more terminology that will come in handy later on. Given a colouring
+c : [Q]n → k, a set C ⊆ k and an interval I = ]u, v[ that is a c-shuffle, we say that I is a
+c-shuffle for the colours in C, or equivalently that I is c-homogeneous for the colours of C,
+if we additionally have c(I) = C.
+2.3
+Preliminaries on Weihrauch reducibility
+We now give a brief introduction to the Weihrauch degrees of problems and the operations
+on them that we will use in the rest of the paper. We stress that here we are able to offer
+but a glimpse of this vast area of research, and we refer to [3] for more details on the topic.
+We deal with partial multifunctions f : ⊆NN ⇒ NN, which we call problems, for short.
+We will most often define problems in terms of their inputs and of the outputs corresponding
+to those inputs. Elements of NN serve as names for the objects we are concerned with, such
+as colourings. Since the encoding of the objects of concern in our paper is trivial, we handle
+this tacitly.
+A partial function F : ⊆ NN → NN is called a realizer for f, which we denote by F ⊢ f, if,
+for every x ∈ dom(f), F(x) ∈ f(x). Given two problems f and g, we say that g is Weihrauch
+reducible to f, and we write g ≤W f, if there are two computable functionals H and K such
+that K⟨FH, id⟩ is a realizer for g whenever F is a realizer for f. We define strong Weihrauch
+
+4
+On the Weihrauch degree of the additive Ramsey theorem
+reducibility similarly: for every two problems f and g, we say that g strongly Weihrauch
+reduces to f, written g ≤sW f, if there are computable functionals H and K such that
+KFH ⊢ g whenever F ⊢ f. We say that two problems f and g are (strongly) Weihrauch
+equivalent if both f ≤W g and g ���W f (respectively f ≤sW g and g ≤sW f). We write this
+≡W (respectively ≡sW).
+We make use of a number of structural operations on problems, which respect the quotient
+to Weihrauch degrees. The first one is the parallel product f × g, which has the power to
+solve an instance of f and and instance of g at the same time. The finite parallelization of
+a problem f, denoted f ∗, has the power to solve an arbitrary finite number of instances of
+f, provided that number is given as part of the input. Finally, the compositional product of
+two problems f and g, denoted f ⋆g, corresponds basically to the most complicated problem
+that can be obtained as a composition of f paired with the identity, a recursive function and
+g paired with identity (that last bit allows to keep track of the initial input when applying
+f).
+Now let us list some of the most important2 problems that we are going to use in the
+rest of the paper.
+CN : ⊆ NN ⇒ N (closed choice on N) is the problem that takes as input an enumeration
+e of a (strict) subset of N and such that, for every n ∈ N, n ∈ CN(e) if and only if
+n ̸∈ ran(e) (where ran(e) is the range of e).
+TCN : ⊆ NN ⇒ N (totalization of closed choice on N) is the problem that takes as input
+an enumeration e of any subset of N (hence now we allow the possibility that ran(e) = N)
+and such that, for every n ∈ N, n ∈ TCN(e) if and only if n ̸∈ ran(e) or ran(e) = N.
+LPO: 2N → {0, 1} (limited principle of omniscience) takes as input any infinite binary
+string p and outputs 0 if and only if p = 0N.
+LPO′ : ⊆ 2N → {0, 1}: takes as input (a code for) an infinite sequence ⟨p0, p1, . . . ⟩ of
+binary strings such that the function p(i) = lims→∞ pi(s) is defined for every i ∈ N, and
+outputs LPO(p).
+Of lesser importance are the following problems:
+Ck (closed choice on k) takes as input an enumeration e of numbers not covering {0, 1, . . . , k−
+1}, and returns a number j < k not covered by e.
+cRT1
+k : kN ⇒ k (Ramsey’s theorem for singletons aka the pigeon hole principle) returns
+some j ∈ k on input p ∈ kN if j occurs infinitely often in p. We point out that cRT1
+k ≡W
+(Ck)′ ≡W RT1
+k (we refer to [4, 7] for details): we prefer to use the “colour version” or
+RT for singletons since it makes many arguments more immediate than the “set version”
+would do.
+cRT1
++ = �
+k>0 cRT1
+k (denoted RT1,+ in [4]) is the disjoint union of the cRT1
+k: it can be
+thought of as a problem taking as input a pair (k, f) where f ∈ N and f : N → k is a
+colouring, and outputting n such that f −1(n) is infinite.
+The definition of LPO′ could have been obtained by composing the one of LPO and the
+definition of jump as given in [3]: we include it for convenience. Intuitively, LPO′ corresponds
+to the power of answering a single binary Σ0
+2-question. In particular, LPO′ is easily seen to
+be (strongly) Weihrauch equivalent to both IsFinite and IsCofinite, the problems accepting
+as input an infinite binary string p and outputting 1 if p contains finitely (respectively,
+cofinitely) many 1s, and 0 otherwise. We will use this fact throughout the paper.
+2 Whereas LPO and CN have been widely studied, TCN is somewhat less known (and does not appear
+in [3]): we refer to [13] for an account of its properties, and to [2] for a deeper study of some principles
+close to it.
+
+A. Pauly, P. Pradic & G. Soldà
+5
+Another problem of combinatorial nature, introduced in [6], will prove to be very useful
+for the rest of the paper.
+▶ Definition 5. ECT is the problem whose instances are pairs (n, f) ∈ N × NN such that
+f : N → n is a colouring of the natural numbers with n colours, and such that, for every
+instance (n, f) and b ∈ N, b ∈ ECT(n, f) if and only if
+∀x > b ∃y > x (f(x) = f(y)).
+Namely, ECT is the problems that, upon being given a function f of the integers with finite
+range, outputs a b such that, after that b, the palette of colours used is constant (hence
+its name, which stands for eventually constant palette tail). We will refer to suitable bs as
+bounds for the function f.
+A very important result concerning ECT and that we will use throughout the paper is
+its equivalence with TC∗
+N.
+▶ Lemma 6 ([6, Theorem 9]). ECT ≡W TC∗
+N
+Another interesting result concerning ECT is the following: if we see it as a statement of
+second-order arithmetic (ECT can be seen as the principle asserting that for every colouring
+of the integers with finitely many colours there is a bound), then ECT and Σ0
+2-IND are
+equivalent over RCA0 (actually, over RCA∗
+0).
+▶ Lemma 7 ([6, Theorem 7]). Over RCA0, ECT and Σ0
+2-IND are equivalent.
+Hence, thanks to the results above, it is clear why TC∗
+N appears as a natural candidate
+to be a “translation” of Σ0
+2-IND in the Weihrauch degrees.
+We end this section with several technical results about Weihrauch degrees.
+Following [13], IsFiniteS : 2N → S is the following problem : for every p ∈ 2N, IsFiniteS(p) =
+⊤ if p contains only ��nitely many occurrences of 1 and IsFiniteS(p) = ⊥ otherwise 3.
+▶ Lemma 8. IsFiniteS ̸≤W ECT
+Proof. Suppose for a contradiction that a reduction exists and is witnessed by functionals
+H and K. We build an instance p of IsFiniteS contradicting this.
+Let us consider the colouring H(0N), and let b0 ∈ ECT(H(0N)) be a bound for it. Since
+IsFiniteS(0N) = ⊤, the outer reduction witness will commit to answering ⊤ after having read
+a sufficiently long prefix of 0N together with b0, say of length n0. Now consider the colouring
+H(0n010N), and a bound b1 > b0 for it. Again by the fact that IsFiniteS(0n010N) = ⊤, there
+is an n1 such that K commits to answering ⊤ after having read the prefix 0n010n1 together
+with b1. We iterate this process indefinitely and obtain an instance p = 0n010n110n21 . . .
+such that IsFiniteS(p) = ⊥.
+However, for the colouring H(p) there must be some bk which is a valid bound, as the
+sequence b0 < b1 < b2 < . . . is unbounded.
+But K will commit to ⊤ upon reading a
+sufficiently long prefix of p together with bk by construction, thereby answering incorrectly.
+◀
+We can now assert that the two main problems that we use as benchmarks in the sequel,
+namely (LPO′)∗ and TC∗
+N, are incomparable in the Weihrauch lattice.
+3 S is the Sierpinski space {⊤, ⊥}, where ⊤ is coded by the binary strings containing at least one 1, and
+⊥ is coded by 0N. IsFiniteS is strictly weaker than IsFinite
+
+6
+On the Weihrauch degree of the additive Ramsey theorem
+▶ Lemma 9. (LPO′)∗ and TC∗
+N are Weihrauch incomparable. Thus (LPO′)∗ <W (LPO′)∗ ×
+TC∗
+N and TC∗
+N <W (LPO′)∗ × TC∗
+N.
+Proof. TC∗
+N ̸≤W (LPO′)∗: to do this, we actually show the stronger result that CN ̸≤W
+(LPO′)∗. Suppose for a contradiction that a reduction exists, as witnessed by the computable
+functionals H and K: this means that, for every instance e of CN, H(e) is an instance of
+(LPO′)∗, and for every solution σ ∈ (LPO′)∗(H(e)), K(e, σ) is a solution to e, i.e. K(e, σ) ∈
+CN(e). We build an instance e of CN contradicting this.
+We start by letting e enumerate the empty set. At a certain stage s, by definition of
+instances of (LPO′)∗, H(e|s) converges to a certain n, the number of applications of LPO′
+that are going to be used in the reduction. Hence, however we continue the construction
+of e, there are at most 2n possible values for (LPO′)∗(H(e)), call them σ0, . . . , σ2n−1. It is
+now simple to diagonalize against all of them, one at a time, as we now explain. We let e
+enumerate the empty set until, for some s0 and i0, K(e|s0, σi0) converges to a certain m0:
+notice that such an i0 has to exist, by our assumption that H and K witness the reduction of
+CN to (LPO′)∗. Then, we let e enumerate m0 at stage s0 +1: this implies that σi0 cannot be
+a valid solution to H(e), otherwise K(e, σi0) would be a solution to e. We then keep letting
+e enumerating {m0} until, for certain s1 and i1, K(e|s1, σi1) converges to m1. We then let
+e enumerate {m0, m1}, and continue the construction in this fashion. After 2n many steps,
+we will have diagonalized against all the σi, thus reaching the desired contradiction.
+(LPO′)∗ ̸≤W TC∗
+N is a consequence of Lemma 8, using the fact that TC∗
+N ≡W ECT (see
+[6]). To see that IsFiniteS ≤W LPO′: given any string p ∈ 2N, we consider the instance
+⟨p0, p1, . . . ⟩ of LPO′ defined as follows: for every i, pi takes value 1 until (and if) the ith
+occurrence of 1 is found in p, after which point it takes value 0. Then, LPO′(⟨p0.p1 . . . ⟩) = 1
+if and only if IsFiniteS(p) = ⊥. Hence, since IsFiniteS ̸≤W ECT, we have in particular that
+(LPO′)∗ ̸≤W TC∗
+N.
+◀
+The second result asserts that the sequential composition of LPO′ × TCN after CN can
+actually be computed by the parallel product of LPO′, TCn
+N and CN. As customary, for every
+problem P we write Pn to mean P × · · · × P
+�
+��
+�
+n times
+. To see that the lemma actually applies to
+CN, we point out that CN ≡W min CN, where min CN is the tightening of CN asking for the
+minimal valid solution.
+▶ Lemma 10. For all a, b ∈ N and every singlevalued problem P :⊆ NN → NN with P ≤W CN,
+it holds that ((LPO′)a × TCb
+N) ⋆ P ≤W (LPO′)a × TCb
+N × P.
+Proof. The proof of TC∗
+N ≡W ECT in [6, Theorem 9] actually shows that TCb
+N ≡W ECTb+1,
+where ECTb+1 is the restriction of ECT to colourings with b + 1 colours. We can thus prove
+the following instead:
+((IsFinite)a × ECTb) ⋆ P ≤W (IsFinite)a × ECTb × P
+We observe that IsFinite(p) = IsFinite(wp) for any w ∈ {0, 1}∗, and if n ∈ ECTb(wp) for some
+w ∈ {0, 1, . . . , b − 1}∗, then n ∈ ECT(p). In other words, both principles have the property
+that adding an arbitrary prefix to an input is unproblematic. As we assume that P ≤W CN,
+there is a finite mindchange computation that solves P.
+In ((IsFinite)a × ECTb) ⋆ P, we can run this finite mindchange computation to obtain the
+inputs for the isFinite and ECTb-instances. Due to the irrelevance of prefixes mentioned
+above, the mindchanges have no problematic impact. Thus, we can actually apply IsFinite
+and ECTb in parallel, which yields the desired reduction to (IsFinite)a × ECTb × P.
+
+A. Pauly, P. Pradic & G. Soldà
+7
+The singlevaluedness of P makes sure that in the parallel execution we get the same
+solution from P as the one used to compute the instances for IsFinite and ECTb.
+◀
+The following shows that the restriction to singlevalued P is necessary in the statement
+of Lemma 10:
+▶ Proposition 11. LPO′ ⋆ C2 ≰W LPO′ × C2
+Proof. The problem LPO′ ⋆ C2 is equivalent to “given p0, p1 ∈ 2N and non-empty A ∈ A(2),
+return (i, isFinite(pi)) for some i ∈ A.”. Let us denote this problem with BI. We will also
+use C2 × IsFinite instead of LPO′ × C2 on the right hand side. We furthermore assume that
+A(2) is represented by ψ : 2N → A(2) where i ∈ ψ(p) iff ∃ℓ p(2ℓ + i) = 1.
+First, we argue that BI ≤W C2 × IsFinite would imply BI ≤sW C2 × IsFinite. Let the outer
+reduction witness be K :⊆ (2N × 2N × 2N) × (2 × 2) → (2 × 2). Note that the inner reduction
+needs to produce inputs to C2 ×IsFinite leading to all four values (i, b) ∈ 2×2 – otherwise, it
+would even show that LPO′⋆C2 ≤W LPO′, which is known to be false for reasons of cardinality.
+Thus, there are prefixes w0, w1 and 0k such that K(w0, w1, 0k, 0, 0) converges. Restricting
+BI to extensions of w0, w1, 0k does not change its strong Weihrauch degree. We then look for
+extensions w1
+0 ≻ w0, w1
+1 ≻ w1 and 0k+ℓ such that K(w1
+0, w1
+1, 0k+ℓ, 0, 1) converges, and do the
+same for the remaining two elements of 2 × 2. By restricting to extensions of those ultimate
+prefixes, we obtain an outer reduction witness that only depends on the 2 × 2-inputs, and
+thus witnesses a strong reduction.
+Next, we disprove BI ≤sW C2 × IsFinite. The outer reduction witness K : 2 × 2 → 2 × 2
+has to be a permutation (as all four values actually occur on the left). The inner reduction
+witness has to map any instance involving 0N as the last component to one involving 0N as
+the first component: If any prefix (w0, w1, 0k) would lead to a C2 × IsFinite-instance where
+the first component is not {0, 1}, then by restricting to the extensions of such an input, we
+would obtain a reduction BI ≤sW IsFinite.
+Let us consider what happens on an input (p, p, 0ω). As above, this gets mapped to some
+(0N, q). We see that the first component of K(i, b) can only depend on b. Moreover, as the
+inner reduction witness cannot map ps with finitely many 1s to qs with infinitely many 1s
+and vice versa, we actually find that the first component of K(i, b) has to be b. Due to
+the symmetry of 0, 1 ∈ 2, this leaves us with two candidates K1, K2 for the outer reduction
+witness we need to consider: K1(i, b) = (b, i) and K2(i, 0) = (0, i), K2(i, 1) = (1, 1 − i).
+Next, we consider inputs of the form (p0, p1, 0N) satisfying that IsFinite(p0) = 1 −
+IsFinite(p1).
+As above, the inner reduction witness with generate some instance (0N, q).
+Depending on q, using either K1 or K2 as outer reduction witness yields either the answers
+(0, 0) and (0, 1); or the answers (1, 0) and (1, 1). However, the correct answers are either
+(1, 0) and (0, 1) or (1, 1) and (0, 0). Thus, both K1 and K2 fail, and we achieved the desired
+contradiction.
+◀
+It will be useful to know the relationship between TCN and cRT1
++. We explore it in the
+following Lemma.
+▶ Proposition 12. cRT1
+2 ≤W TCN, but cRT1
+3 ≰W TCN. In particular, cRT1
++ ̸≤W TCN.
+Proof. For the reduction cRT1
+2 ≤W TCN, just list 2n in the TCN-instance when the n-th 1
+appears in the RT1
+2-instance, and list 2n + 1 when the n-th 0 appears in the RT1
+2-instance.
+When TCN returns some n ∈ N, return n mod 2 as answer to cRT1
+2.
+To see that cRT1
+3 ̸≤W TCN, it is enough to notice that TCN ≤W SRT2
+2 (see [18, Proposition
+7.24]), since it is known that RT1
+k+1 ̸≤W SRT2
+k (see [4, Corollary 6.6]).
+◀
+
+8
+On the Weihrauch degree of the additive Ramsey theorem
+2.4
+Green theory
+Green theory is concerned with analysing the structure of ideals of finite semigroups, be
+they one-sided on the left or right or even two-sided. This gives rise to a rich structure
+to otherwise rather inscrutable algebraic properties of finite semigroups. We will need only
+a few related results, all of them relying on the definition of the Green preorders and of
+idempotents (recall that an element s of a semigroup is idempotent when ss = s).
+▶ Definition 13. For a semigroup (S, ·), define the Green preorders as follows:
+•
+s ≤R t
+if and only if
+s = t or s ∈ tS = {ta : a ∈ S}
+(suffix order)
+•
+s ≤L t
+if and only if
+s = t or s ∈ St = {at : a ∈ S}
+(prefix order)
+•
+s ≤H t
+if and only if
+s ≤R t and s ≤L t
+•
+s ≤J t
+if and only if
+s ≤R t or s ≤L t or s ∈ StS = {atb : (a, b) ∈ S2}
+(infix order)
+The associated equivalence relations are written R, L, H, J ; their equivalence classes are
+called respectively R, L, H, and J -classes.
+We conclude this section reporting, without proof, three technical lemmas that will be
+needed in Section 4 and 5. Although not proved in second-order arithmetic originally, it is
+clear that their proofs goes through in RCA0: besides straightforward algebraic manipula-
+tions, they only rely on the existence, for each finite semigroup (S, ·), of an index n ∈ N
+such that sn is idempotent for any s ∈ S.
+▶ Lemma 14 ([14, Proposition A.2.4]). If (S, ·) is a finite semigroup, H ⊆ S an H-class,
+and some a, b ∈ H satisfy a · b ∈ H then for some e ∈ H we know that (H, ·, e) is a group.
+▶ Lemma 15 ([14, Corollary A.2.6]). For any pair of elements x, y ∈ S of a finite semigroup,
+if we have x ≤R y and x, y J -equivalent, then x and y are also R-equivalent.
+▶ Lemma 16 ([14, Corollary A.2.6]). For every finite semigroup S and s, t ∈ S, s ≤L t and
+s R t implies s H t.
+3
+The shuffle principle and related problems
+3.1
+The shuffle principle in reverse mathematics
+We start by giving a proof4 of the shuffle principle in RCA0 + Σ0
+2-IND, since, in a way, it
+gives a clearer picture of some properties of shuffles that we use in the rest of the paper.
+▶ Lemma 17. RCA0 + Σ0
+2-IND ⊢ Shuffle
+Proof. Let c : Q → n be a colouring of the rationals with n colours. For any natural number
+k, consider the following Σ0
+2 formula ϕ(k): “there exists a finite set L ⊆ n of cardinality k
+and there exist u, v ∈ Q with u < v such that c(w) ∈ L for every w ∈ ]u, v[”. Since ϕ(n) is
+true, it follows from the Σ0
+2 minimization principle that there exists a minimal k such that
+ϕ(k) holds. Consider u, v ∈ Q and the set of colours L corresponding to this minimal k. We
+now only need to show that ]u, v[ is a c-shuffle to conclude.
+Let a = c(x) for some x ∈ ]u, v[.
+We need to prove that a occurs densely in ]u, v[.
+Consider arbitrary x, y ∈ ]u, v[ with x < y.
+We are done if we show that there exists
+4 From Leszek A. Kołodziejczyk, personal communication.
+
+A. Pauly, P. Pradic & G. Soldà
+9
+some w ∈ ]x, y[ with c(w) = a. So, suppose that there is no such w. By bounded Σ0
+1-
+comprehension, there exists a finite set L′ ⊂ n consisting of exactly those b ∈ n which occur
+as values of c
+��
+]x,y[. Clearly, ϕ(|L′|) holds. However, L′ ⊆ L, and by assumption a /∈ L′, so
+|L′| < k, contradicting the choice of k as the minimal number such that ϕ(k) holds.
+◀
+The proof above shows an important feature of shuffles: given a certain interval ]u, v[, any
+of its subintervals having the fewest colours is a shuffle. Interestingly, the above implication
+reverses, so we have the following equivalence.
+▶ Theorem 18. Over RCA0, Shuffle is equivalent to Σ0
+2-IND.
+We do not offer a proof of the reversal here; such a proof can easily be done by taking
+inspiration from the argument we give for Lemma 27.
+With this equivalence in mind, we now introduce Weihrauch problems corresponding to
+Shuffle, beginning with the stronger one.
+▶ Definition 19. We regard Shuffle as the problem with instances (k, c) ∈ N × NN such that
+c : Q → k is a colouring of the rationals with k colours, and such that, for every instance
+(k, c), for every pair (u, v) ∈ [Q]2 and for every C ⊆ k, (u, v, C) ∈ Shuffle(k, c) if and only
+if ]u, v[ is a c-shuffle for the colours in C.
+Note that the output of Shuffle contains two components that cannot be easily computed
+from one another. It is very natural to split the principles into several problems, depending
+on the type of solution that we want to be given: one problem will output the colours of a
+shuffle, whereas another will output the interval. As we will see, the strength of these two
+versions of the same principle have very different uniform strength.
+▶ Definition 20. iShuffle (“i” for “interval”) is the same problem as Shuffle save for the fact
+that a valid output only contains the interval ]u, v[ which is a c-shuffle. Complementarily,
+cShuffle (“c” for “colour”) is the problem that only outputs a possible set of colours taken by
+a c-shuffle.
+We will first start analysing the weaker problems cShuffle and iShuffle and show they are
+respectively equivalent to (LPO′)∗ and TC∗
+N. This will also imply that Shuffle is stronger
+than (LPO′)∗ × TC∗
+N, but the converse will require an entirely distinct proof.
+3.2
+Weihrauch complexity of the weaker shuffle problems
+We first provide a classification of cShuffle, by gathering a few lemmas. The first also applies
+to iShuffle and Shuffle.
+▶ Lemma 21. cShuffle × cShuffle ≤W cShuffle, iShuffle × iShuffle ≤W iShuffle and Shuffle ×
+Shuffle ≤W Shuffle. Hence, cShuffle∗ ≡W cShuffle, iShuffle∗ ≡W iShuffle and Shuffle∗ ≡W
+Shuffle.
+Proof. Consider the pairing of the two input colourings. To give more details, let (n0, f0)
+and (n1, f1) be instances of Shuffle. Let us fix a computable bijection ⟨·, ·⟩ : n0 × n1 → n0n1
+and define the colouring f : Q → n0n1 by f(x) = ⟨f0(x), f1(x)⟩ for every x ∈ Q. Hence,
+(n0n1, f) is a valid instance of Shuffle.
+Let C ∈ cShuffle(n0n1, f): this means that there is an interval I that is a f-shuffle for
+the colours of C. For i < 2, let Ci := {j : ∃c ∈ C(j = πi(j))}, where πi is the projection on
+the ith component. Then, Ci ∈ cShuffle(nifi), as witnessed by the interval I.
+
+10
+On the Weihrauch degree of the additive Ramsey theorem
+With the same reasoning, if I ∈ iShuffle(n0n1, f), then also I ∈ iShuffle(n0, f0) and
+I ∈ iShuffle(n1, f1). Finally, if (I, C) ∈ Shuffle(n0n1, f), then (I, C0) ∈ Shuffle(n0, f0) and
+(I, C1) ∈ Shuffle(n1, f1).
+To conclude Shuffle∗ ≡W Shuffle from Shuffle×Shuffle ≤W Shuffle, we just need to observe
+that Shuffle has a computable instance; likewise for cShuffle and iShuffle.
+◀
+▶ Lemma 22. LPO′ ≤W cShuffle
+Proof. We will prove that IsFinite ≤sW cShuffle. Let p ∈ 2N be an infinite binary string,
+we define a colouring c : Q → 2 of the rationals by setting c( n
+m) = p(m) for n ∈ Z,m ∈ N
+with gcd(n, m) = 1. If 1 ∈ C ∈ cShuffle(c), then, since density implies infinity, p must have
+infinitely many occurrences of 1. On the other hand, if for infinitely many n p(n) = 1, then
+all the reduced fractions of the form a
+n are coloured 1 by c, which implies that the colour 1
+occurs densely in every interval. Hence, 1 ∈ C if and only if 1 appeared in p infinitely often,
+which proves the claim.
+◀
+Putting Lemmas 21 and 22 together, we see that we can solve n instances of LPO′ by
+using cShuffle on a 2n-colouring. For the reversal, we incur an exponential increase in the
+parameter as well:
+▶ Lemma 23. Let cShufflen be the restriction of cShuffle to the instances of the form (n, c).
+Then, cShufflen ≤W (LPO′)2n−1
+Proof. We actually show that cShufflen ≤W IsFinite2n−1. Let (n, c) be an instance of cShuffle.
+The idea is that we will use one instance of IsFinite for every non-empty subset C of the set
+of colours n, in order to determine for which such Cs there exists an interval IC such that
+c(IC) = C. We will then prove that any ⊆-minimal such C is a solution for (n, c).
+Let Ci, for i < 2n − 1, be an enumeration of the non-empty subsets of n. Let Ij be
+an enumeration of the open intervals of Q, and let qh be an enumeration of Q. For every
+i < 2n − 1, we build an instance pi of IsFinite in stages in parallel. At every stage s, for
+every component i < 2n − 1, there will be a “current interval” Ijis and a “current point” qhis.
+We start the construction by setting the current interval to I0 and the current point to q0
+for every component i.
+For every component i, at stage s we do the following:
+if qhis ̸∈ Ijis or if c(qhis) ∈ Ci, we set Iji
+s+1 = Ijis and qhi
+s+1 = qhs+1. Moreover, we set
+pi(s) = 0. In practice, this means that if the colour of the current point is in Ci, or if
+the current point is not in the current interval, no special action is required, and we can
+move to consider the next point.
+If instead qhis ∈ Ijis and c(qhis) ̸∈ Ci, we set Iji
+s+1 = Ijs+1 and qhi
+s+1 = q0. Moreover, we
+set pi(s) = 1. In practice, this means that if the current point is in the current interval
+and its colour is not a colour of Ci, then, we need to move to consider the next interval
+in the list, and therefore we reset the current point to the first point in the enumeration.
+Moreover, we record this event by letting pi(s) take value 1.
+We iterate the construction for every s ∈ N. After infinitely many steps, we obtain an in-
+stance ⟨p0, p1, . . . , p2n−2⟩ of IsFinite2n−1. Let σ ∈ 22n−1 be such that σ ∈ IsFinite2n−1(⟨p0, p1, . . . , p2n−2⟩).
+To find a set of colours C for which there is a c-shuffle, we proceed s follows. We start
+checking σ(i) for i such that Ci is a singleton: if, for any such i, σ(i) = 1, it means that
+the corresponding pi has only finitely many 1s, which implies that the second case in the
+construction was triggered only finitely many times. Hence, there is a stage s such that, for
+
+A. Pauly, P. Pradic & G. Soldà
+11
+every t > s, Ijis = Iji
+t. This means that Ijis is c-homogeneous, and thus, in particular, a
+c-shuffle. Hence, Ci is a valid solution.
+If instead for all is such that Ci is a singleton σ(i) = 0: then, we know that no interval I
+is c-monochromatic, otherwise we would be in the previous case. We move to consider the
+is such that |Ci| = 2. Suppose that for one such i, σ(i) = 1: again, this means that, for a
+sufficiently large stage s, the current interval Ijis is such that, for every q ∈ Ijis, c(q) ∈ Ci,
+since the second case in the construction is triggered only finitely many times. But since
+we know that there are no c-monochromatic intervals, the two colours of Ci occur densely
+in Ijis, which then is a c-shuffle for the colours in Ci. Hence, any Ci such that σ(i) = 1 is a
+valid solution for c.
+This argument can be iterated for every number of colours.
+Since, by the theory, a
+c-shuffle exists, at least one of the pi instances above contains only finitely many 1s. To
+compute a solution to c, it is thus sufficient to look for the minimal k such that, for some i,
+σ(i) = 1 and |Ci| = k, and output Ci.
+◀
+Putting the previous lemmas together, we have the following:
+▶ Theorem 24. (LPO′)∗ ≡W cShuffle
+Proof. (LPO′)∗ ≤W cShuffle is given by Lemmas 21 and 22. For the other direction, notice
+that cShuffle ≡W
+�
+n∈N cShufflen. The result then follows from Lemma 23.
+◀
+While Theorem 24 tells us that for any finite number of parallel LPO′-instances can be
+reduced to cShuffle for m-colourings for a suitable choice of m, and vice versa, a sufficiently
+large number of LPO′-instances can solve cShuffle for m-colourings, both directions of our
+proof involved an exponential increase in the parameter. Before moving on to iShuffle, we
+thus raise the open question of whether this gap can be narrowed:
+▶ Question 25. What is the relationship between (LPO′)n and cShufflem for individual
+n, m ∈ N?
+▶ Lemma 26. Let iShufflen be the restriction of iShuffle to the instances of the form (n, c).
+For n ≥ 1, it holds that iShufflen ≤sW TCn−1
+N
+.
+Proof. Fix an enumeration Ij of the intervals of Q, an enumeration qh of Q, a computable
+bijection ⟨·, ·⟩: N × N → N, and let (n, c) be an instance of iShufflen.
+The idea of the reduction is the following: with the first instance en−1 of TCN, we look
+for an interval Ij on which c takes only n − 1 colours: if no such interval exists, then this
+means that every colour is dense in every interval, and so every Ij is a valid solution to
+c. Hence, we can suppose that such an interval is eventually found: we will then use the
+second instance en−2 of TCN to look for a subinterval of Ij where c takes only n − 2 values.
+Again, we can suppose that such an interval is found. We proceed like this for n − 1 steps,
+so that in the end the last instance e1 of TCN is used to find an interval I′ inside an interval
+I on which we know that at most two colours appear: again, we look for c-monochromatic
+intervals: if we do not find any, then I′ is already a c-shuffle, whereas if we do find one, then
+that interval is now a solution to c, since c-monochromatic intervals are trivially c-shuffles..
+Although not apparent in the sketch given above, an important part of the proof is that
+the n − 1 searches we described can be performed in parallel: the fact that this can be
+accomplished relies on the fact that any subinterval of a shuffle is a shuffle. More formally,
+we proceed as follows: we define n − 1 instances e1, . . . , en−1 of TCN as follows. For every
+
+12
+On the Weihrauch degree of the additive Ramsey theorem
+stage s, every instance ei will have a “current interval” Ijis and a “current point” qhis and a
+“current list of colours” Lkis. We start the construction by the setting the current interval
+equal to I0, the current point equal to q0 and the current list of points equal to ∅ for every
+i.
+At stage s, there are two cases:
+if, for every i, qhis ̸∈ Ijis or |Lkis ∪ {c(qhis)}| ≤ i, we set Iji
+s+1 = Ijis, qhi
+s+1 = qhis+1 and
+Lki
+s+1 = Lkis ∪ {c(qhis)}. Moreover, we let every ei enumerate every number of the form
+⟨s, a⟩, for every a ∈ N, except for ⟨s, ji
+s⟩. We then move to stage s + 1.
+In practice, this means that if the set of colours of the points of the current interval seen
+so far does not have cardinality larger than i, no particular action is required, and we
+can move to check the next point on the list.
+otherwise: let i′ be maximal such that qhis ∈ Ijis and |Lkis ∪ {c(qhis)}| > i. Then, for
+every i > i′ we proceed as in the previous case (i.e., the current interval, current point,
+current list of colours and enumeration are defined as above). For the other components,
+we proceed as follows: we first look for the minimal ℓ > ji′
+s such that Iℓ ⊆ Iji′+1
+s
+(if
+i′ = n − 1, just pick ℓ = jn−1
+s
++ 1). Then, for every i ≤ i′, we set Iji
+s+1 = Iℓ, qhi
+s+1 = q0
+and Lki
+s+1 = ∅. Moreover, we let ei enumerate every number of the form ⟨t, a⟩ with t < s
+that had not been enumerated so far, and also every number of the form ⟨s, a⟩, with the
+exception of ⟨s, ji
+s⟩. We then move to stage s + 1.
+In practice, this means that if, for a certain component i′, we found that the current
+interval has too many colours, then, for all the components i ≤ i′, we move to consider
+intervals strictly contained in the current interval of component i′.
+We iterate the procedure for every s ∈ N, thus obtaining the TCn−1
+N
+-instance ⟨e1, . . . , en−1⟩.
+Let σ ∈ Nn−1 be such that σ ∈ TCn−1
+N
+(⟨e1, . . . , en−1⟩). Then, we look for the minimal
+i such that Iπ2(σ(i)) ⊆ Iπ2(σ(i+1)) ⊆ · · · ⊆ Iπ2(σ(n−1)) (by πi we denote the projection on
+the ith component, so ⟨π1(x), π2(x)⟩ = x)). We claim that Iπ2(σ(i)) is a c-shuffle, which is
+sufficient to conclude that iShufflen ≤sW TCn−1
+N
+.
+We now prove the claim. First, suppose that en−1 enumerates all of N. Then, the second
+case of the construction was triggered infinitely many times with i′ = n − 1: hence, no
+interval contains just n − 1 colours, and so, as we said at the start of the proof, this means
+that every interval is a c-shuffle. In particular, this imples that Iπ2(σ(i)) is a valid solution.
+Hence we can suppose that en−1 does not enumerate all of N.
+Next, we notice that for every m > 1, if em enumerates all of N, the so does em−1, by
+inspecting the second case of the construction. Let m be minimal such that em does not
+enumerate all of N. For such an m, it is easy to see that Iπ2(σ(m)) is a valid solution to c:
+indeed, we know from the construction that c takes m colours on Ipi2(σ(m)), and that for no
+interval contained in Iπ2(σ(m)) c takes m−1 colours, which means that Iπ2(σ(m)) is a c-shuffle.
+Moreover, it is easy to see that Iπ2(σ(m)) ⊆ Iπ2(σ(m+1)) ⊆ . . . Iπ2(σ(n−1)), which implies that
+i ≤ m. Since every subinterval of a c-shuffle is a c-shuffle, Iπ2(σ(i)) is a valid solution to c,
+as we wanted.
+◀
+▶ Lemma 27. Let ECTn be the restriction of ECT to the instances of the form (n, f). It
+holds that ECTn ≤sW iShufflen.
+Proof. Let (n, f) be an instance of ECTn.
+We define c: Q → n by c( a
+b ) = f(b) where
+gcd(a, b) = 1. Hence, all the points of the same denominator have the same colour according
+to c. Let ( u
+k , v
+ℓ ) ∈ iShufflen(n, c). Let b be such that 1
+b < v
+ℓ − u
+k . We claim that b is a bound
+for f. Suppose not, then there is a colour i < n and a number x ∈ N such that x > b and
+f(x) = i, but for no y > x it holds that f(y) = i. Hence, all the reduced of the form w
+x are
+
+A. Pauly, P. Pradic & G. Soldà
+13
+given colour i, but i does not appear densely often in any interval of Q. But by choice of b,
+there is a z ∈ Z such that z
+b ∈
+� u
+k , v
+ℓ
+�
+, which is a contradiction. Hence b is a bound for f.
+◀
+Putting things together, we finally have a characterization of iShuffle. We even get a
+precise characterization for each fixed number of colours.
+▶ Theorem 28. We have the Weihrauch equivalence
+ECTn ≡W iShufflen ≡W TCn−1
+N
+whence
+ECT ≡W iShuffle ≡W TC∗
+N
+Proof. We get TCn−1
+N
+≤W ECTn by inspecting the second half of [6, Theorem 9]. Then
+Lemma 27 gives us ECTn ≤W iShufflen. Lemma 26 closes the cycle by establishing iShufflen ≡W
+TCn−1
+N
+.
+◀
+3.3
+The full shuffle problem
+The main result of this section is that Shuffle ≡W TC∗
+N × (LPO′)∗, which will be proved
+in Theorem 31. For one direction, we merely need to combine our results for the weaker
+versions:
+▶ Lemma 29. TC∗
+N × (LPO′)∗ ≤W Shuffle
+Proof. From Theorem 24 and Theorem 28, we have that TC∗
+N × (LPO′)∗ ≤W iShuffle ×
+cShuffle, and since clearly iShuffle ≤W Shuffle and cShuffle ≤W Shuffle, by Lemma 21 we
+have that TC∗
+N × (LPO′)∗ ≤W Shuffle.
+◀
+For the other direction, again, we want to be precise as to the number of TCN- and
+(LPO′)-instances we use to solve an instance of Shuffle. Note that we will use a far larger
+number of TCN-instances to obtain a suitable interval than we used in Lemma 26.
+▶ Lemma 30. Let Shufflen be the restriction of Shuffle to the instances of the form (n, c).
+Then, Shufflen ≤W (TCN × LPO′)2n−1
+Proof. Let (n, c) be an instance of Shuffle. The idea of the proof of Shufflen ≤W (TCN ×
+LPO′)2n−1 is, in essence, to combine the proofs of Lemma 26 and of Lemma 23: we want to
+use TCN to find a candidate interval for a certain subset C of n, and on the side we use LPO′
+(or equivalently, IsFinite) to check for every such set C whether a c-shuffle for the colours of
+C actually exists. The main difficulty with the idea described above is that the two proofs
+must be intertwined, in order to be able to find both a c-shuffle and the set of colours that
+appears on it.
+We proceed as follows: let Ci be an enumeration of the non-empty subsets of n. Moreover,
+let us fix some computable enumeration Ij of the intervals of Q, some computable enumer-
+ation qh of the points of Q, and some computable bijection ⟨·, ·⟩: N × N → N. For every
+Ci, we will define an instance ⟨pi, ei⟩ of IsFinite × TCN in stages as follows: at every stage s,
+for every index i, there will be a “current interval” Ijis and a “current point” qhis. We begin
+stage 0 by setting the current interval to I0 and the current point to q0 for every index i.
+At stage s, for every component i, there are two cases:
+if qhis ̸∈ Ijis or if c(qhis) ∈ Ci, we set Iji
+s+1 = Ijis and qhi
+s+1 = qhi
+s+1. Moreover, we set
+pi(s) = 0 and we let ei enumerate all the integers of the form ⟨s, a⟩, except ⟨s, ji
+s+1⟩. We
+then move to stage s + 1.
+In plain words, for every component i, we check if the colour of the current point is in
+Ci, or if the current point is not in the current interval: if this happens, no special action
+is required.
+
+14
+On the Weihrauch degree of the additive Ramsey theorem
+If instead qhis ∈ Ijis and c(qhis) ̸∈ Ci, we set Iji
+s+1 = Ijis+1 and qhi
+s+1 = q0. Moreover, we
+set pi(s) = 1, and we let ei enumerate all the numbers of the form ⟨t, a⟩, for t < s, that
+had not been enumerated at a previous stage, and also all the numbers of the form ⟨s, a⟩,
+with the exception of ⟨s, ji
+s+1⟩. We then move to stage s + 1.
+In plain words: if we find that for some component i the colour of the current point is
+not in Ci, then, from the next stage, we start considering another interval, namely the
+next one in the fixed enumeration. We then reset the current point to q0 (so that all
+rationals are checked again), and we record the event by letting pi(s) = 1 and changing
+the form of the points that ei is enumerating.
+We iterate the procedure for every integer s. Let σ ∈ (2 × N)2n−1 be such that
+σ ∈ (IsFinite × TCN)2n−1(⟨⟨p1, e1⟩ . . . , ⟨p2n−1, e2n−1⟩)⟩
+Let k be the minimal cardinality of a subset Ci ⊆ n such that IsFinite(pi) = 1: notice that
+such a k must exist, because c-shuffle exist. Then, we claim that the corresponding Iπ2(σ(i))
+is a c-shuffle (by πi we denote the projection on the ith component, so ⟨π1(x), π2(x)⟩ = x)).
+If we do this, it immediately follows that Shuffle ≤W ((LPO′) × TCN)2n−1.
+Hence, all that is left to be done is to prove the claim. By the fact that IsFinite(pi) = 1, we
+know that the second case of the construction is triggered only finitely many times. Hence,
+ei does not enumerate all of N, and so Iπ2(σ(i)) is an interval containing only colours from
+Ci. Moreover, by the minimality of |Ci|, we know that no subinterval of Iπ2(σ(i)) contains
+fewer colours, which proves that Iπ2(σ(i)) is a c-shuffle.
+◀
+Putting the previous results together, we obtain the following.
+▶ Theorem 31. Shuffle ≡W TC∗
+N × (LPO′)∗
+3.4
+The (η)1
+<∞-problem
+A weakening of the shuffle principle was studied in [8] under the name (η)1
+<∞. The principle
+(η)1
+<∞ asserts that for any colouring of Q in finitely many colours, some colour will be dense
+somewhere. We formalize it here as follows:
+▶ Definition 32. The principle (η)1
+<∞ takes as input a pair (k, α) where k ∈ N and α : Q → k
+is a colouring, and returns an interval I and a colour n < k such that α−1(n) is dense in
+I. The principle i(η)1
+<∞ returns only the interval I, c(η)1
+<∞ only the dense colour n. Let
+(c(η)1
+<∞)k be the restriction of c(η)1
+<∞ to k-colourings.
+An important aspect of the definition above to notice is that we require a bound on the
+number of colours used to be declared in the instance of (η)1
+<∞.
+While (η)1
+<∞ also exhibits the pattern that we can neither compute a suitable interval
+from knowing the dense colour nor vice versa, we shall see that as far as the Weihrauch
+degree is concerned, finding the interval is as hard as finding both interval and colour. Our
+proof does not preserve the number of colours though.
+▶ Proposition 33. (η)1
+<∞ ≡W i(η)1
+<∞ ≡W TC∗
+N ≡W iShuffle
+Proof. Taking into account Theorem 28, it suffices for us to show that (η)1
+<∞ ≤W iShuffle
+and that ECT ≤W i(η)1
+<∞. For (η)1
+<∞ ≤W iShuffle we observe that an interval which is
+a shuffle not only has a dense colour in it, but every colour that appears is dense. Thus,
+we return the interval obtained from iShuffle on the same colouring, together with the first
+colour we spot in that interval.
+
+A. Pauly, P. Pradic & G. Soldà
+15
+It remains to prove that ECT ≤W i(η)1
+<∞. Given a k-colouring c of N, we will compute
+a 2k-colouring α of Q. We view the 2k-colouring as a colouring by subsets of k, i.e. each
+rational gets assigned a set of the original colours. To determine whether the n-th rational qn
+should be assigned the colour j < k, we consider the number mn,j = |{s | s ≤ n ∧ c(s) = j}|
+of prior ocurrences of the colour j in c. If the integer part of qn ∗ 2mn,j is odd, qn is assigned
+colour j, otherwise not.
+If j appears only finitely many times in c, then mn,j is eventually constant, and the
+distribution of j in α follows (with finitely many exceptions) the pattern of alternating
+intervals of width 2−mn,j. This ensures that none of the 2k-many colours for α can be dense
+on an interval wider than 2−mn,j. Subsequently, we find that the width of the interval having
+a dense colour returned by i(η)1
+<∞ provide a suitable bound to return for ECT.
+◀
+The Proposition above implies that c(η)1
+<∞ has to be weaker than (LPO′)∗, since it is
+immediate to see that it is computed by both (η)1
+<∞ and cShuffle. We now give more bounds
+on its strength.
+▶ Lemma 34. (c(η)1
+<∞)k+1 ≤W cRT1
+k+1 × (c(η)1
+<∞)k
+Proof. Fix some enumeration (In)n∈N of all rational intervals.
+The forwards reduction
+witness is constructed as follows.
+We keep track of an interval index n and a colour c,
+starting with n = 0 and c = 0. We keep writing the current value of c to the input of
+cRT1
+k+1, and we construct a colouring β : Q → {0, 1, . . . , k − 1} by scaling the colouring α
+restricting to In up to Q, while excluding c and subtracting 1 from every colour d > c. The
+fact that we may have already assigned β-colours to finitely many points in a different way
+before is immaterial.
+If we ever find a rational q ∈ In with α(q) = c < k, we increment c. If we find q ∈ In
+with α(q) = c = k, we set c = 0 and increment n. In particular, we stick with any particular
+In until we have found points of all different colours inside it.
+The backwards reduction witness receives two colours, c ∈ {0, 1, . . . , k} and d ∈ {0, 1, . . . , k−
+1}. If d < c, it returns d. If d ≥ c, it returns d + 1.
+To see that the reduction works correctly, first consider the case where every colour
+is dense everywhere.
+In this case, everything is a correct answer, and the reduction is
+trivially correct. Otherwise, there has to be some interval In and some colour c such that
+α−1(c) ∩ In = ∅. In this case, our updating of n and c will eventually stabilize at such a
+pair. The answer we will receive from cRT1
+k+1 is c. Apart from finitely many points, β will
+be look like the restriction of α to In with c skipped. Thus, any colour d which is dense
+somewhere for β will be dense somewhere inside In for α if d < c, or if d ≥ c, then d + 1 will
+be dense. Thus, the reduction works.
+◀
+▶ Corollary 35.
+(c(η)1
+<∞)k ≤W cRT1
+k × cRT1
+k−1 × . . . × cRT1
+2
+≤W (cRT1
+2)k−1 × (cRT1
+2)k−2 × . . . (cRT)1
+2
+≡W (cRT1
+2)k(k−1)/2
+These bounds allow us to characterize the stregth of c(η)1
+<∞.
+▶ Corollary 36. c(η)1
+<∞ ≡W cRT1
++
+Proof. The direction c(η)1
+<∞ ≤W cRT1
++ is provided by Corollary 35. For the other direction
+we show cRT1
+k ≤W (c(η)1
+<∞)k. Fix a computable bijection ν : N → Q. Given a colouring
+
+16
+On the Weihrauch degree of the additive Ramsey theorem
+f : N → k as input for cRT1
+k, we define the colour αf : Q → k by αf(q) = f(ν−1(q)). Clearly,
+any colour appearing somewhere dense in αf must have appeared infinitely often in f.
+◀
+Our result that c(η)1
+<∞ ≡W cRT1
++ stands in contrast to the reverse mathematics results
+obtained in [8]. In reverse mathematics, RT1
+N is equivalent to BΣ0
+2 [10], yet [8, Theorem 3.5]
+shows that BΣ0
+2 does not imply (η)1
+<∞ over RCA0.
+4
+ARTQ and related problems
+We now analyse the logical strength of the principle ARTQ. As in the case of Shuffle, we
+start with a proof of ARTQ in RCA0 + Σ0
+2-IND. This will give us enough insights to assess
+the strength of the corresponding Weihrauch problems.
+4.1
+Additive Ramsey over Q in reverse mathematics
+As a preliminary step, we figure out the strength of ORTQ, the ordered Ramsey theorem over
+Q. It is readily provable from RCA0 and is thus much weaker than most other principles we
+analyse. We can be a bit more precise by considering RCA∗
+0 which is basically the weakening
+of RCA0 where induction is restricted to ∆0
+1 formulas (see [17, Definition X.4.1] for a nice
+formal definition).
+▶ Lemma 37. RCA∗
+0 ⊢ RCA0 ⇔ ORTQ
+We now show that the shuffle principle is equivalent to ARTQ. So overall, much like the
+Ramsey-like theorems of [11], they are equivalent to Σ0
+2-induction.
+▶ Lemma 38. RCA0 + Shuffle ⊢ ARTQ. Hence, RCA0 + Σ0
+2-IND ⊢ ARTQ.
+Proof. Fix a finite semigroup (S, ·) and an additive colouring c : [Q]2 → S. Say a colour
+s ∈ S occurs in X ⊆ Q if there exists (x, y) ∈ [X]2 such that c(x, y) = s.
+We proceed in two stages: first, we find an interval ]u, v[ such that all colours occurring
+in ]u, v[ are J -equivalent to one another. Then we find a subinterval of ]u, v[ partitioned
+into finitely many dense homogeneous sets. For the first step, we apply the following lemma
+to obtain a subinterval I1 = ]u, v[ of Q where all colours lie in a single J -class.
+▶ Lemma 39. For every additive colouring c, there exists (u, v) ∈ [Q]2 such that all colours
+of c
+��
+]u,v[ are J -equivalent to one another.
+Proof. If we post-compose c with a map taking a semigroup element to its J -class, we get
+an ordered colouring. Applying ORTQ yields a suitable interval.
+◀
+Moving on to stage two of the proof, we want to look for a subinterval of I1 partitioned
+into finitely many dense homogeneous sets. To this end, define a colouring γ : I1 → S2 by
+setting γ(z) = (c(u, z), c(z, v)).
+By Shuffle, there exist x, y ∈ I1 with x < y such that ]x, y[ is a γ-shuffle. For l, r ∈ S,
+define Hl,r : = γ−1({(l, r)}) ⊆ ]x, y[; note that this is a set by bounded recursive compre-
+hension. Clearly, all Hl,r are either empty or dense in ]x, y[, and moreover ]x, y[ = �
+l,r Hl,r.
+Since there are finitely many pairs (l, r), all we have to prove is that each non-empty Hl,r is
+homogeneous for c.
+Let s = c(w, z) such that w, z ∈ Hl,r with w < z. By additivity of c and the definition
+of Hl,r,
+s · r = c(w, z) · c(z, v) = c(w, v) = r.
+(1)
+
+A. Pauly, P. Pradic & G. Soldà
+17
+In particular r ≤R s. But we also have r J s, which gives r R s by Lemma 15. This shows
+that all the colours occurring in Hl,r are R-equivalent to one another. A dual argument
+shows that they are all L-equivalent, so they are all H-equivalent.
+The assumptions of
+Lemma 14 are satisfied, so their H-class is actually a group.
+All that remains to be proved is that any colour s occurring in Hl,r is actually the
+(necessarily unique) idempotent of this H-class.
+Since r R s, there exists a such that
+s = r ·a. But then by (1), s·s = s·r ·a = r ·a = s, so s is necessarily the idempotent. Thus,
+all sets Hl,r are homogeneous and we are done.
+◀
+We conclude this section by showing that the implication proved in the Lemma above
+reverses., thus giving the precise strength of ARTQ over RCA0.
+▶ Theorem 40. RCA0 + ARTQ ⊢ Shuffle. Hence, RCA0 ⊢ ARTQ ↔ Σ0
+2-IND.
+Proof. Let f : Q → n be a colouring of the rationals. Let (Sn, ·) be the finite semigroup
+defined by Sn = n and a · b = a for every a, b ∈ Sn. Define the colouring c: [Q]2 → Sn
+by setting c(x, y) = f(x) for every x, y ∈ Q. Since for every x < y < z, c(x, z) = f(x) =
+c(x, y) · c(y, z), c is additive.
+By additive Ramsey, there exists ]u, v[ which is c-densely
+homogeneous and thus a f-shuffle.
+◀
+4.2
+Weihrauch complexity of additive Ramsey
+We now start the analysis of ARTQ in the context of Weihrauch reducibility. We will mostly
+summarize results, relying on the intuitions we built up so far. First off, we determine the
+Weihrauch degree of the ordered Ramsey theorem over Q.
+▶ Theorem 41. Let ORTQ be the problem whose instances are ordered colourings c : [Q]2 →
+P, for some finite poset (P, ≺), and whose possible outputs on input c are intervals on which
+c is constant. We have that ORTQ ≡W LPO∗.
+Proof. LPO∗ ≤sW ORTQ: let ⟨n, p0, . . . , pn−1⟩ be an instance of LPO∗. Let (P, ≺) be the
+poset such that P = 2n, the set of subsets of n, and ≺ = ⊃, i.e. ≺ is reverse inclusion.
+We define an ordered colouring c : [Q]2 → P in stages by deciding, at stage s, the colour
+of all the pairs of points (x, y) ∈ [Q]2 such that |x − y| > 2−s.
+At stage 0, we set c(x, y) = ∅ for every (x, y) ∈ [Q]2 such that |x−y| > 1. At stage s > 0,
+we check pi
+��
+s for every i < n (i.e., for every i, we check the sequence pi up to pi(s − 1)), and
+for every (x, y) ∈ [Q]2 with 2−s+1 ≥ |x − y| > 2−s, we let
+c(x, y) = {i < n : ∃t < s(pi(t) = 1)}.
+It is easily seen that c defined as above is an ordered colouring: if x ≤ x′ < y′ ≤ y′, then
+|x′ − y′| ≤ |x − y|, which means that to determine the colour of (x′, y′) we need to examine
+a longer initial segment of the pis. Let I ∈ ORTQ(P, c), and let ℓ ∈ N be least such that
+the length of I is larger that 2−ℓ: since I is c-homogeneous, we know that for every i < n,
+∃t(pi(t) = 1) ⇔ ∃t < ℓ(pi(t) = 1). Hence, for every pair of points (x, y) ∈ [I]2, the colour of
+c(x, y) is exactly the set of i such that LPO(pi) = 1.
+ORTQ ≤W LPO∗: Let (P, c) be an instance of ORTQ, for some finite poset (P, ≺P ). Let
+<L be a linear extension of ≺P , and notice that c : Q → (P, <L) is still an ordered colouring.
+Let r0 <L r1 <L · · · <L r|P |−1 be the elements of P. The idea of the proof is to have one
+instance of LPO per element of P, and to check in parallel the intervals of the rationals to
+see if they are c-homogeneous for the corresponding element of P. Anyway, one has to be
+careful as to how these intervals are chosen: to give an exampe, if we find that a certain
+
+18
+On the Weihrauch degree of the additive Ramsey theorem
+interval I is not c-homogeneous for the <L-maximal element r|P |−1, because we found, say,
+x < y such that c(x, y) ̸= r|P |−1, then not only do we flag the corresponding instance of
+LPO by letting it contain a 1, but we also restrict the research of all the other components
+so that they only look at intervals contained in ]x, y[. By proceeding similarly for all the
+components, since c is ordered, we are sure that we will eventually find a c-homogeneous
+interval.
+We define the |P| instances p0, p1, . . . , p|P |−1 of LPO in stages as follows. Let an be an
+enumeration of the ordered pairs of rationals, i.e. an enumeration of [Q]2, with infinitely many
+repetitions. At every stage s, some components i will be “active”, whereas the remaining
+components will be “inactive”: if a component i is inactive, it can never again become active.
+Moreover, at every stage s, there is a “current pair” ans and a “current interval” ams (for this
+proof, it is practical to see ordered pairs of rational as both pairs and as denoting extrema
+of an open interval). We begin stage 0, by putting the current pair and the current interval
+equal to a0. Moreover, every component is set to be active.
+At stage s, for every inactive component j < |P|, we set pj(s) = 1. For every active
+component i, there are two cases:
+if, for every active component i, c(ans) ≥L ri, then we look for the smallest ℓ > ns such
+that aℓ ⊆ ams (i.e., we look for a pair of points contained in the current interval), and
+set ans+1 = aℓ, and ams+1 = ams. We set pi(s) = 0 and no component is set to inactive.
+We then move to stage s + 1.
+suppose instead there is an active component i such that c(ans) <L ri: let i be the
+minimal such i, then we set every j ≥ i to inactive (the ones that were already inactive
+remain so) and we let pj(s) = 1. We then let ams+1 = ans, and we look for the least
+ℓ > ns such that aℓ ⊂ ans: we set ans+1 = aℓ, and we set pk(s) = 0 for every active
+component k < |P|. We then move to stage s + 1.
+We iterate the procedure above for every integer s.
+Let σ ∈ 2|P | be such that σ ∈ LPO∗(⟨|P|, p0, . . . , p|P |−1⟩). Notice that σ(0) = 0, since no
+pair of points can attain colour <L-below r0. Moreover, notice that σ(i) = 0 if and only if
+the component i was never set inactive. Hence, let i be maximal such that σ(i) = 0, and let
+t be a state such all components j > i have been set inactive by step t. Hence, after step t,
+the current interval I never changes, and thus we eventually check the colour of all the pairs
+in that interval. Since the second case of the construction is never triggered, it follows that
+I is a c-homogeneous interval. Hence, in order to find it, we know we just have to repeat
+the construction above until all the components of index larger than i are set inactive. This
+proves that ORTQ ≤W LPO∗.
+◀
+Now let us discuss Weihrauch problems corresponding to ARTQ.
+▶ Definition 42. Regard ARTQ as the following Weihrauch problem: the instances are pairs
+(S, c) where S is a finite semigroup and c : [Q]2 → S is an additive colouring of [Q]2, and
+such that, for every C ⊆ S and every interval I of Q, (I, C) ∈ ARTQ if and only if I is
+c-densely homogeneous for the colours of C.
+Similarly to what we did in Definition 20, we also introduce the problems cARTQ and iARTQ
+that only return the set of colours and the interval respectively.
+We start by noticing that the proof of Theorem 40 can be readily adapted to show the
+following.
+▶ Lemma 43.
+cShuffle ≤sW cARTQ, hence (LPO′)∗ ≤W cARTQ.
+
+A. Pauly, P. Pradic & G. Soldà
+19
+iShuffle ≤sW iARTQ, hence TC∗
+N ≤W iARTQ.
+Shuffle ≤sW ARTQ, hence (LPO′)∗ × TC∗
+N ≤W ARTQ.
+The rest of the section is devoted to find upper bounds for cARTQ, iARTQ and ARTQ.
+The first step to take is a careful analysis of the proof of Lemma 38. For an additive colouring
+c: [Q]2 → S, the proof can be summarized as follows:
+we start with an application of ORTQ to find an interval ]u, v[ such that all the colours
+of c
+��
+]u,v[ are all J -equivalent (Lemma 39).
+define the colouring γ : Q → S2 and apply Shuffle to it, thus obtaining the interval ]x, y[.
+the rest of the proof consists simply in showing that ]x, y[ is a c-densely homogeneous
+interval.
+Hence, from the uniform point of view, this shows that ARTQ can be computed via a
+composition of Shuffle and ORTQ. Whence the next theorem.
+▶ Theorem 44.
+cARTQ ≤W (LPO′)∗ × LPO∗, therefore cARTQ ≡W (LPO′)∗.
+iARTQ ≤W TC∗
+N × LPO∗, therefore iARTQ ≡W TC∗
+N.
+ARTQ ≤W (LPO′)∗ × TC∗
+N × LPO∗, therefore ARTQ ≡W (LPO′)∗ × TC∗
+N.
+Proof. The three results are all proved in a similar manner. We recall that LPO∗ ≤W CN
+and observe that LPO∗ is single-valued. This enables us to use Lemma 10 with LPO∗ in
+place of P.
+For x ∈ {c, i, s} and every n ∈ N, let xARTQ,n be the restriction of xARTQ to instances of
+the form (S, c) with S of cardinality n. Hence, by the considerations preceding the statement
+of the theorem in the body of the paper, we have the following facts:
+cARTQ,n ≤W cShufflen2 ∗ ORTQ, hence, by Lemma 23 and Theorem 41, we have that
+cARTQ,n ≤W (LPO′)2n2−1∗LPO∗. By Lemma 10, we have that cARTQ,n ≤W (LPO′)2n2−1×
+LPO∗, from which the claim follows.
+iARTQ,n ≤W iShufflen2 ∗ ORTQ, hence, by Lemma 26 and Theorem 41, we have that
+iARTQ,n ≤W TCn2−1
+N
+∗ LPO∗. By Lemma 10, we have that iARTQ,n ≤W TCn2−1
+N
+× LPO∗,
+from which the claim follows.
+ARTQ,n ≤W Shufflen2 ∗ ORTQ, hence, by Lemma 30 and Lemma 10, we have that
+ARTQ,n ≤W (LPO′ × TCN)2n2−1 ∗ LPO∗.
+By Lemma 10, we have that ARTQ,n ≤W
+(LPO′ × TCN)2n2−1 × LPO∗, from which the claim follows.
+◀
+5
+ARTN and ORTN
+We finally turn to the case of the additive and ordered theorems over N and prove Theorem 3.
+We obtain results which are completely analogous to the case of Q when it comes to the
+additive Ramsey theorem. However, in contrast to Theorem 41, the ordered Ramsey theorem
+for N exhibits the same behaviour as the additive Ramsey theorem.
+That the principles ORTN and ARTN are equivalent to Σ0
+2-induction
+was established
+in [11], so we only focus on the analysis of the Weihrauch degrees below. We first start by
+defining properly the principles involved, and then we give the proof that TC∗
+N, (LPO′)∗ or
+their product reduces to them. We then give the converse reductions, first for the principles
+pertaining to the ordered colourings, and then we handle the additive colourings.
+The
+proof for the ordered colouring is a simple elaboration on [11, Lemma 4.3]. For the additive
+colouring, formally the corresponding statement in that paper, [11, Proposition 4.1], depends
+
+20
+On the Weihrauch degree of the additive Ramsey theorem
+on the ordered version in a way that would translate to a composition in the setting of
+Weihrauch degrees. It turns out that we can avoid invoking the composition by carefully
+interleaving the two steps in our analysis.
+5.1
+Definitions
+We have already covered the principles ORTN and ARTN in Section 2. The corresponding
+Weihrauch problems are relatively clear: given a colouring as input, as well as the finite
+semigroup or finite ordered structure, output an infinite homogeneous set. The principles
+cORTN and cARTN instead only output a possible colour for an infinite homogeneous set –
+much like in the case for Q. However, the principles iORTN and iARTN will require some
+more attention; now it is rather meaningless to ask for a containing interval. Nevertheless,
+the analogous principle will also output some information regarding the possible location of
+an homogeneous set, without giving away a whole set or a candidate colour, so we keep a
+similar naming convention.
+▶ Definition 45. Define the following Weihrauch problems:
+ORTN takes as input a finite poset (P, ⪯P ) and a right-ordered colouring c : [N]2 → P,
+and outputs an infinite c-homogeneous set ⊆ N.
+ARTN takes as input a finite semigroup S and an additive colouring c : [N]2 → S, and
+outputs an infinite c-homogeneous set ⊆ N.
+cORTN takes as input a finite poset (P, ⪯P ) and a right-ordered colouring c : [N]2 → P,
+and outputs a colour p ∈ P such that there exists an infinite c-homogeneous set ⊆ N with
+colour p.
+cARTN takes as input a finite semigroup S and an additive colouring c : [N]2 → S, and
+outputs a colour s ∈ S such that there exists an infinite c-homogeneous set ⊆ N with
+colour s.
+iORTN takes as input a finite poset (P, ⪯P ) and a right-ordered colouring c : [N]2 → P,
+and outputs a n0 ∈ N such that there is an infinite c-homogeneous set X ⊆ N with two
+elements ≤ n0.
+iARTN takes as input a finite semigroup S and an additive colouring c : [N]2 → S, and
+outputs a n0 ∈ N such that there is an infinite c-homogeneous set X ⊆ N with two
+elements ≤ n0.
+5.2
+Reversals
+▶ Lemma 46. We have ECT ≤sW iORTN and ECT ≤sW iARTN.
+Proof. Let f : N → k be a would-be instance of ECT. Then one may define the colouring
+˜f : [N]2 → P(k) by setting a ∈ ˜f(n, m) if and only if there is n′ with n ≤ n′ ≤ m and
+f(n′) = a.
+This colouring is both additive for the semigroup (P(k), ∪) and ordered by
+⊆, and can be fed to either iORTN or iARTN. Let n0 be such that there is an infinite ˜f-
+homogeneous set with first two elements k0 < k1 ≤ n0. Clearly, every colour occuring in f
+after n0 needs to occur in ˜f(k0, k1); so n0 is a solution of the given instance for ECT.
+◀
+▶ Lemma 47. cORT∗
+N ≡W cORTN and cART∗
+N ≡W cARTN
+Proof. The non-trivial reductions are easily made by amalgamating finite sequences of col-
+ouring via a pointwise product, which will always still carry an additive or ordered struc-
+ture.
+◀
+
+A. Pauly, P. Pradic & G. Soldà
+21
+▶ Lemma 48. LPO′ ≤sW cORTN and LPO′ ≤sW cARTN.
+Proof. We use IsFinite in place of LPO′. We start with an input f : N → 2 for IsFinite. We
+compute ˜f : [N]2 → 2 where ˜f(n, m) = 1 iff 1 ∈ f −1([n, m]). This yields an additive and
+ordered colouring. The colour of any given ˜f-homogeneous set indicates if f has infinitely
+many ones or not, thus answering IsFinite for f.
+◀
+5.3
+Reducing the ordered Ramsey theorem over N to (LPO′)∗ and ECT
+We now explain how to bound the Weihrauch degree of ORTN and its weakenings. To do so,
+it will be helpful to consider a construction approximating would-be homogeneous sets for a
+given right-ordered colouring c : [N]2 → P and a target colour p ∈ P. With these parameters,
+we build a recursive sequence of finite sets Y (p) : N → Pfin(N) meant to approximate a p-
+homogeneous set (we shall simply write Y instead of Y (p) when p may be inferred from
+context). If the construction succeeds, lim sup(Y ) will be an infinite homogeneous set with
+colour p, otherwise lim sup(Y ) will be finite. But the important aspect will be that a fixed
+number of calls to (LPO′)∗ will let us know if the construction was successful or not, while
+ECT can indicate after which indices n we shall have Yn ⊆ lim sup(Y ) when it succeeds.
+Now let us describe this construction for a fixed c and p. We begin with Y0 = ∅ and will
+maintain the invariant that max(Yn) < n and for every (k, k′) ∈ [Yn]2, c(k, k′) = p. Then,
+for Yn+1, we have several possibilities;
+If min(Yn) exists and for any min(Yn) ≤ k < n, we have that p ≺P c(k, n), we set
+Yn+1 = ∅ and say that the construction was injured at stage n.
+Otherwise, if we have some k′ < n such that c(k′, n) = p and, for every k ∈ Yn, k < k′
+and c(k, k′) = p, then we set Yn+1 = Yn ∪ {k} and say that the construction progressed
+at stage n.
+Otherwise, set Yn+1 = Yn and say that the construction stagnated.
+Clearly, we can also define recursive sequences injury(p)
+n
+: N → 2 and progress(p)
+n
+: N → 2
+that witness whether the construction was injured or progressed, and we have that lim sup(Y )
+is infinite if and only if injury contains finitely many 1 and progress contains infinitely many
+ones. lim sup(Y ) is moreover always c-homogeneous with colour p.
+▶ Lemma 49. For any ordered colouring c, there is p such that lim sup(Y ) is infinite
+Proof. The suitable p may be found as follows: say that a colour p occurs after n in c if there
+is k > m ≥ n with c(m, k) = p. There is a n0 such that every colour occuring after n0 in c
+occurs arbitrarily far. For the ⪯P -maximal such colour occuring after n0, the construction
+above will succeed with no injuries after stage n0 and infinitely many progressing steps (this
+is exactly the same argument as for [11, Lemma 4.3]).
+◀
+▶ Lemma 50. We have that cORTN ≤sW (LPO′)∗.
+Proof. Given an input colouring c, compute in parallell all injury(p) and progress(p) for every
+colour p and feed each sequence to an instance of LPO′. By Lemma 49, there is going to be
+some p for which there is going to be finitey many injuries and infinitely many progressing
+steps, and that p is the colour of some homogeneous set.
+◀
+▶ Lemma 51. We have that ORTN ≤W (LPO′)∗ × ECT.
+Proof. Given an input colouring c, compute in parallell all injury(p) and progress(p) for
+every colour p and feed each sequence to an instance of LPO′ and all injury(p) to ECT. As
+
+22
+On the Weihrauch degree of the additive Ramsey theorem
+before, use LPO′ to find out some p for which the construction succeed. For that p, ECT
+will yield some n0 such that injury(p)
+n
+= 0 for every n ≥ n0, so in particular, lim sup(Y (p)) =
+�
+n≥n0 Y (p)
+n
+, which is computable from n0.
+◀
+▶ Lemma 52. We have that iORTN ≤sW ECT.
+Proof. Given an input colouring c, consider for every colour p the sequence u(p) : N →
+{0, 1, 2} defined by u(p)
+n
+= min(3, |Yn|). Clearly it is computable from c. Applying ECT we
+get some np such that
+either there are infinitely many injuries after np
+or u(p)
+k
+= u(p)
+np for every k ≥ np
+By Lemma 49, we even know there is a p0 such that lim sup(Y (p0)) is infinite; additionally we
+defined u in such a way that necessarily, np0 bounds two elements of lim sup(Y (p0)) because
+we shall have u(p0)
+k
+= 2 for every k ≥ np0. So we may simply take the maximum of all np to
+solve our instance of iORTN.
+◀
+This concludes our analysis of the ordered Ramsey theorem.
+5.4
+Reducing the additive Ramsey theorem over N to (LPO′)∗ and ECT
+We now turn to ARTN. The basic idea is that, given an additive colouring c, it is useful
+to define the composite colouring L ◦ c, with L being a map from a finite semigroup to its
+L-classes. ≤R then induces a right-ordered structure on the colouring. Constructing a L ◦ c
+homogeneous set X such that we additionally have that c(min X, x) = c(min X, y) for every
+x, y ∈ X \ {min X} ensures that X is c-homogeneous by Lemma 16. So we will give a recipe
+to construct exactly such an approximation, similarly to what we have done in the previous
+section.
+So this time around, assume a semigroup S and a colouring c : [N]2 → S to be fixed.
+For every s ∈ S, we shall define a recursive sequence of sets Y (s) : N → Pfin(N) (we omit
+the superscript when clear from context) such that max(Yn) < n and Yn \ {min(Yn)} be
+homogeneous, with, if Yn ̸= ∅, c(min(Yn), k) = s for k ∈ Yn \ {min(Yn)}.
+For n = 0, we define Y0 = ∅. For Yn+1, we have a couple of options:
+If Yn is empty and there is k < n such that c(k, n) = s and there is no k ≤ k′ < n′ ≤ n
+with c(k′, n′) <R s, then set Yn+1 = {k} for the minimal such k and say that the
+construction (re)starts.
+If Yn is non-empty and there is some k with min(Yn) ≤ k < n with c(k, n) <R s, set
+Yn+1 = ∅ and say that the construction was injured at stage n.
+Otherwise, if Yn is non-empty and we have some min(Yn) < k < n with c(min(Yn), k) = s
+and c(k, n) R s, set Yn+1 = Yn ∪ {k} and say that the construction progresses.
+Otherwise, set Yn+1 = Yn and say that the construction stagnates.
+We can define auxiliary binary sequences injury(s) and progress(s) that witness the rel-
+evant events, and an infinite homogeneous subset will be built as long as we have finitely
+many injuries and infinite progress.
+▶ Lemma 53. If injury(s) has finitely many 1s and progress(s) has infinitely many 1s, then
+X = lim sup(Y )\{min(lim sup(Y ))} is a c-homogeneous infinite set. Furthermore the colour
+of X is computable from s.
+
+A. Pauly, P. Pradic & G. Soldà
+23
+Proof. That the condition is sufficient for X to be infinite is obvious; it only remains to show
+it is homogeneous. Note that all elements in lim sup(Y ) are necessarily R-equivalent to one
+another. Call y0 = min(lim sup(Y )). For (l, m) ∈ [X]2, we necessarily have s ≤L c(l, m). So
+by Lemma 16, we necessarily have c(l, m) H s. Additionally, we know they belong to a H-
+class which is a group, so by algrebraic manipulations, we have that [X]2 is monochromatic
+and the corresponding colour is the neutral element of that group.
+◀
+▶ Lemma 54. There is some s such that injury(s) has finitely many 1s and progress(s) has
+infinitely many 1s.
+Proof. Consider, much as we did in the proof of Lemma 49, a n0 such that all R-classes
+occurring after n0 occur arbitrarily far. Consider a minimal such R-class R. For the minimal
+k0 such that no R-class strictly lower than R occurs after k0, there is some s such that the set
+{n | c(k0, n) = s} is infinite; but, by construction, this set is exactly lim sup(Y (s))\{k0}.
+◀
+With these two lemmas in hand, it is easy then to carry out a similar analysis as in
+the last subsection. We do not expand the proof, which are extremely similar, but only
+summarize the results.
+▶ Lemma 55. We have the following reductions:
+cARTN ≤sW (LPO′)∗
+ARTN ≤W (LPO′)∗ × ECT
+iARTN ≤sW ECT
+This concludes the analysis of ARTN and its natural weakenings.
+6
+How the colours are coded
+All principles we have studied that receive as input a colouring of some sort also receive
+explicit finite information about the finite set/finite poset/finite semigroup of colours. This
+is not the approach we could have taken: in the case of a plain set of colours, the colouring
+itself contains the information on how many colours it is using. In the cases where the
+colours carry additional structure, this could have been provided via the atomic diagram of
+the structure. This would lead to the requirement that only finitely colours are used to be
+a mere promise.
+We will first demonstrate the connection between the two versions on a simple example,
+namely cRT1
++. Let us denote with cRT1
+N the principle that takes as input a colours α : N → N
+such that the range of α is finite, and outputs some n ∈ N such that α−1(n) is infinite.
+▶ Proposition 56. cRT1
++ ⋆ CN ≡W cRT1
+N
+Proof. Instead of cRT1
++ ⋆CN ≤W cRT1
+N we show that cRT1
++ ⋆Bound ≤W cRT1
+N, where Bound
+receives as input an enumeration of a finite initial segment of N, and outputs an upper bound
+for it. Here is how we produce the input to cRT1
+N given an input A to Bound and a sequence
+(ki, αi)i∈N of partial inputs to cRT1
++: We search for some ki0 to be defined, and then start
+copying αi0 until i0 gets enumerated into A (if this never happens, αi0 is total and becomes
+the input to cRT1
+N. Then we search for some i1 > i0 such that ki1 is defined, and then
+continue to produce the colouring αi1 + k0; either forever or until i1 gets enumerated into
+A. We repeat this process until some iℓ is reached which is never enumerated into A (this
+has to happen).
+
+24
+On the Weihrauch degree of the additive Ramsey theorem
+Given a colour c that appears infinitely often in the resulting colouring, we can retrace
+our steps and identify what iℓ was. We can then un-shift c to obtain a colour appearing
+infinitely often in αiℓ, and thereby answer cRT1
++ ⋆ CN.
+For the converse direction, we observe that CN can compute from a colouring α : N → N
+with finite range some k ∈ N such that α is a k-colouring.
+◀
+The very same relationship holds for all our principles, i.e. the Weihrauch degree of the
+version without finite information on the colours is just the composition of the usual version
+with CN. The core idea, as in the proposition above, is that we can always start over by
+moving to a fresh finite set of colours. For the interval versions we may have to do a little
+bit more work to encode the CN-output by ensuring that all “large” intervals can never be
+a valid answer.
+To see that this observation already fully characterizes the Weihrauch reductions and
+non-reductions between the usual and the relaxed principles, the notion of a (closed) fractal
+from [1, 12] is useful.
+▶ Definition 57. A Weihrauch degree f is called a fractal, if there is some F :⊆ NN ⇒ NN
+with f ≡W F such that for any w ∈ N∗ either wNN ∩ dom(F) = ∅ or F|wNN ≡W f. If we
+can chose F to be total, we call the Weihrauch degree a closed fractal.
+If f is a fractal and f ≤W
+�
+i∈N gi, then there has to be some n ∈ N with f ≤W gn.
+If f is a closed fractal and f ≤W g ⋆ CN, then already f ≤W g. Of our principles, the
+versions with a fixed number of colours are closed fractals, the versions with a given-but-
+not-fixed number of colours are not fractals at all, and the versions without explicit colour
+information are fractals, but not closed fractals. From this, it follows that the versions with
+no explicit colour information are never Weihrauch equivalent to our studied principles, and
+that versions without explicit colour information are equivalent to one-another if and only
+if their counterparts with explicit colour information are equivalent.
+7
+Conclusion and future work
+Summary
+We have analysed the strength of an additive Ramseyan theorem over the ra-
+tionals from the point of view of reverse mathematics and found it to be equivalent to Σ0
+2-
+induction, and then refined that analysis to a Weihrauch equivalence with TC∗
+N × (LPO′)∗.
+We have also shown that the problem decomposes nicely: we get the distinct complexities
+(LPO′)∗ or TC∗
+N if we only require either the set of colours or the location of the homogeneous
+set respectively. The same holds true for another equally and arguably more fundamental
+shuffle principle, as well as the additive Ramsey theorem over N that was already studied
+from the point of view of reverse mathematics in [11].
+Perpectives
+It would be interesting to study further mathematical theorems that are
+known to be equivalent to Σ0
+2-IND in reverse mathematics: this can be considered to contrib-
+ute to the larger endeavour of studying principles already analyzed in reverse mathematics
+in the framework of the Weihrauch degrees. In the particular case of Σ0
+2-IND, it can be
+interesting to see which degrees are necessary for such an analysis. We refer to [4] for more
+on this topic, and for a more comprehensive study of Ramsey’s theorem in the Weihrauch
+degrees.
+
+A. Pauly, P. Pradic & G. Soldà
+25
+Acknowledgements
+The second author warmly thanks Leszek Kołodziejczyk for the proof of Lemma 17 as well as
+Henryk Michalewski and Michał Skrzypczak for numerous discussions on a related project.
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+GPT as Knowledge Worker:
+A Zero-Shot Evaluation of (AI)CPA Capabilities
+Jillian Bommaritoa, Michael J Bommarito IIa,b,c,d, Jessica Katza, Daniel Martin Katza,b,c,d
+a273 Ventures LLC
+bIllinois Tech - Chicago Kent College of Law
+cBucerius Law School
+dCodeX - The Stanford Center for Legal Informatics
+Abstract
+The global economy is increasingly dependent on knowledge workers to meet the needs of public and private organizations. While
+there is no single definition of knowledge work, organizations and industry groups still attempt to measure individuals’ capability
+to engage in it. The most comprehensive assessment of capability readiness for professional knowledge workers is the Uniform
+CPA Examination developed by the American Institute of Certified Public Accountants (AICPA). In this paper, we experimentally
+evaluate OpenAI’s text-davinci-003 and prior versions of GPT on both a sample Regulation (REG) exam and an assessment of
+over 200 multiple-choice questions based on the AICPA Blueprints for legal, financial, accounting, technology, and ethical tasks.
+First, we find that text-davinci-003 achieves a correct rate of 14.4% on a sample REG exam section, significantly underperforming
+human capabilities on quantitative reasoning in zero-shot prompts. Second, text-davinci-003 appears to be approaching human-
+level performance on the Remembering & Understanding and Application skill levels in the Exam absent calculation. For best
+prompt and parameters, the model answers 57.6% of questions correctly, significantly better than the 25% guessing rate, and its
+top two answers are correct 82.1% of the time, indicating strong non-entailment. Finally, we find that recent generations of GPT-3
+demonstrate material improvements on this assessment, rising from 30% for text-davinci-001 to 57% for text-davinci-003. These
+findings strongly suggest that large language models have the potential to transform the quality and efficiency of future knowledge
+work.
+Keywords: knowledge work, artificial intelligence, natural language processing, accounting, finance, law
+Introduction
+Knowledge work is an increasingly important segment of
+the global economy, with qualified professionals providing ser-
+vices in areas such as law, finance, accounting, economics, and
+technology.
+Leading management theorists began exploring
+definitions of “knowledge workers” and approaches for their
+training nearly seven decades ago [1, 2, 3]. Since then, the per-
+centage of the population that “thinks for a living” has grown
+dramatically. As of 2021, the Big 4 - Deloitte, EY, PWC, and
+KPMG - alone employ over one million people [4]; some def-
+initions of knowledge work suggest that the true number of
+knowledge workers is in the hundreds of millions or even bil-
+lions [5].
+As their roles and activities may generate substantial value
+- and liability - many organizations require these knowledge
+workers to demonstrate their preparedness through comprehen-
+sive assessments, such as the so-called CPA, CFA, or Bar ex-
+ams. While there is no universally-accepted definition of knowl-
+edge work [6], public accounting is a multidisciplinary practice
+that requires legal, financial, accounting, auditing, technology,
+and ethical knowledge and skills - all domains clearly within
+Email address: [email protected] (Jillian Bommarito)
+the scope of knowledge work. As the test used to assess the
+readiness of candidates for this profession, the American In-
+stitute of Certified Public Accountants (AICPA) Uniform CPA
+Examination (“CPA Exam” or “Exam”) is the most compre-
+hensive, well-known assessment of knowledge work readiness
+[7]. As compared to other assessments or examinations, the
+CPA Exam is broader, more practice-based, and more regu-
+larly updated to meet the changing landscape. This trend is
+perhaps best demonstrated by the fact that the commercial or-
+ganizations most associated with the AICPA - the Big 4 - have
+accumulated practically every type of knowledge work under
+their umbrella, including even cybersecurity and traditional le-
+gal services [8, 9, 10].
+The AICPA and the National Association of State Boards
+of Accountancy have undertaken a joint effort to ensure that the
+CPA licensure model reflects the “rapidly changing skills and
+competencies the practice of accounting requires today and will
+require in the future” [11]. The Exam is produced by the AICPA
+based on input from stakeholders in the professional services
+industry, academia, and governmental agencies. The Exam has
+been continually updated to meet changing regulations, stan-
+dards, technology, and market expectations for over 100 years
+[7, 12]. While the Exam continues to evolve [12, 13], it was his-
+torically adapted from the best-known educational framework,
+Preprint submitted to arxiv
+January 12, 2023
+arXiv:2301.04408v1  [cs.CL]  11 Jan 2023
+
+Bloom’s cognitive taxonomy [2], to organize the assessment of
+practical, professional requirements into four skill levels [14].
+Though the exam will undergo significant structural changes in
+2024, the current implementation of the exam has been divided
+into four sections: Auditing and Attestation (AUD), Business
+Environment and Concepts (BEC), Financial Accounting and
+Reporting (FAR), and Regulation (REG). These four sections
+cover concepts, laws, rules, and relationships in legal, finan-
+cial, accounting, and technology domains, common denomina-
+tors among many knowledge professions.1
+Previous decades of research into artificial intelligence (AI)
+have not yielded general models capable of performing knowl-
+edge work. While point solutions in many legal, financial, or
+accounting domains have shown value or reached adoption, there
+has been no demonstration of AI that can span multiple task
+types in professional services. This gap can likely be attributed
+to multiple reasons, including the breadth and depth of knowl-
+edge required to be indexed and recalled, as well as the com-
+plexity of translating this knowledge into work product in the
+context of realistic client engagements. To make matters more
+difficult, professional services like accounting, finance, and law
+also often require a combination of quantitative and qualitative
+skills.
+Recent research has, however, shown potential to address
+at least some of these capability gaps.
+Advances in natural
+language processing (NLP), machine learning (ML), and com-
+puting over the last decade have produced material improve-
+ments in state-of-the-art performance on linguistic tasks that
+require deeper semantic understanding or feature more com-
+plex syntax [15] [16] [17]. More importantly, some types of
+models have begun to demonstrate the ability to address dra-
+matically different task types, sometimes even in zero-shot use
+cases where there is no additional fine-tuning or customization.
+While neural network research is not new [18] [19], the rate
+of progress has increased dramatically since 2013, and, in par-
+ticular, transformer-based architectures [20] have been shown
+to produce previously-unseen capabilities to generalize across
+tasks [21] [22] [23] [24] [25].
+The most accessible and well-known of these transformer-
+based models is OpenAI’s family of large language models known
+as Generative Pre-trained Transformer or “GPT” [22] [26]. The
+latest versions of GPT, often referred to as GPT-3 or GPT-3.5,
+are proprietary large language models, and these models are
+only available to OpenAI customers. One benefit of this ap-
+proach is that it provides an important layer of legal and ethical
+moderation, as well as simplifying the user experience, such
+as by preprocessing input text or images. As of this publica-
+tion, the OpenAI provides API endpoints for text completion,
+code completion, image generation, and embedding generation
+tasks. OpenAI has also recently unveiled ChatGPT, a public-
+facing “chatbot” built on GPT-3.5, which reportedly generated
+over 1M user sign-ups within just a few days of release.
+As GPT-3 and its derivatives are proprietary machine learn-
+ing models in production within a reinforcement learning plat-
+form, we cannot precisely describe them. However, based on
+1Interested readers should review Table 4 for the list of all concept areas.
+GPT-3’s original publication in July 2020 and subsequent ma-
+terial, these models are likely derived from an autoregressive
+language model with 175 billion parameters, 96 layers, and a
+batch size of 3.2M. OpenAI has launched or published a num-
+ber of GPT-3 derivative models, most notably InstructGPT-3
+and Codex 12B, which are colloquially referred to as GPT-3.5.
+The most advanced model in production in its API is text-
+davinci-003, an improvement on text-davinci-002, which is an
+InstructGPT model based on code-davinci-002, a base model for
+pure code-completion tasks, per OpenAI documentation. Our
+results in this paper are primarily based on text-davinci-003, as
+detailed in Section d, though we also include results from older
+models for comparison and forecasting.
+While text-davinci-003 and ChatGPT have demonstrated
+state-of-the-art performance on a wide range of tasks in zero-
+shot and few-shot contexts, there was previously little reason
+to believe that these models could perform even reasonably
+well in general assessments across the domains of finance, law,
+and accounting.
+However, in recent prior work on the Bar
+Exam [27], the authors have shown that text-davinci-003 could
+achieve near-parity with human test-takers in two of seven sec-
+tions of the Multistate Bar Exam (MBE); more strikingly, generation-
+over-generation model performance suggests that an LLM like
+GPT-3.5 may be capable of passing the Bar Exam in the near
+future.
+While the Bar Exam offered one measure of performance
+for GPT-3.5, it is arguably not the ideal instrument to evalu-
+ate readiness for multidisciplinary knowledge work. As noted,
+the CPA Exam requires a wider range of knowledge, includ-
+ing not only law, but also finance, accounting, technology, and
+ethics. Therefore, in order to evaluate whether and how current
+state-of-the-art models in AI might be applied to knowledge
+work, we experimentally evaluate the performance of “GPT as
+knowledge worker” through the skills and concepts outlined in
+the CPA Exam. Our analysis suggests both areas where GPT-
+3.5 may be useful today and areas where substantial research
+and development is still required.
+AICPA Exam
+The Uniform CPA Examination is a modern, computerized
+assessment based on psychometric and statistical techniques.
+While prior paper-based generations of the Exam might have
+been compared to traditional linear exams, the current Exam is
+a dynamic, adaptive exam [28], best compared to exams like
+the current GRE or GMAT. Linear exams present the test-taker
+with a preset sequence of test questions, while dynamic exams
+adapt to each test-taker in response to the answers provided in
+prior questions.
+Section
+Student Pass Rate
+AUD
+48.7%
+BEC
+59.7%
+FAR
+44.9%
+REG
+61.1%
+Table 1: Passage rates of students in 2022 as reported by the AICPA [29].
+2
+
+Skill Level
+Description
+Evaluation
+The examination or assessment of problems, and use of judgment to draw con-
+clusions.
+Analysis
+The examination and study of the interrelationships of separate areas in order to
+identify causes and find evidence to support inferences.
+Application
+The use or demonstration of knowledge, concepts, or techniques.
+Remembering &
+Understanding
+The perception and comprehension of the significance of an area utilizing
+knowledge gained.
+Table 2: AICPA Uniform CPA Examination Skill Levels
+Skill
+Area
+Content
+Task
+Remembering
+&
+Understanding
+Internal
+Controls
+Sarbanes-Oxley
+Act of 2002
+Identify and define key corporate governance
+provisions of the Sarbanes-Oxley Act of 2002.
+Application
+Internal
+Controls
+Sarbanes-Oxley
+Act of 2002
+Identify regulatory deficiencies within an entity
+by using the requirements associated with the
+Sarbanes-Oxley Act of 2002.
+Table 3: Example AICPA Uniform CPA Examination Tasks
+Auditing and Attestation (AUD)
+Ethics, Professional Responsibilities and General Principles
+Assessing Risk and Developing a Planned Response
+Performing Further Procedures and Obtaining Evidence
+Forming Conclusions and Reporting
+Business Environment and Concepts (BEC)
+Enterprise Risk Management, Internal Controls and Business Processes
+Economics
+Financial Management
+Information Technology
+Operations Management
+Financial Accounting and Reporting (FAR)
+Conceptual Framework, Standard-Setting and Financial Reporting
+Select Financial Statement Accounts
+Select Transactions
+State and Local Governments
+Regulation (REG)
+Ethics, Professional Responsibilities and Federal Tax Procedures
+Business Law
+Federal Taxation of Property Transactions
+Federal Taxation of Individuals
+Federal Taxation of Entities
+Table 4: Uniform CPA Examination Blueprints - Content Areas
+3
+
+The Examination is divided into four sections that test-takers
+sit for independently: Auditing and Attestation (AUD), Busi-
+ness Environment and Concepts (BEC), Financial Accounting
+and Reporting (FAR), and Regulation (REG). Each section of
+the Exam is divided up into at least four testlets that feature
+scenarios, multiple choice questions, calculated amounts, short
+answer, and related evidence and research material. The pas-
+sage rates of Exam sections are presented in Table 1; the AICPA
+does not publish statistics related to per-question or per-section
+test-taker accuracy.
+By its very design, the Exam is meant to be a practical as-
+sessment of real-world tasks and requisite skills [11, 28]. It
+rigorously assesses candidates on their readiness across a broad
+range of concepts and skill levels progressing through (i) Re-
+membering & Understanding, (ii) Application, (iii) Analysis,
+and (iv) Evaluation.
+The overall design of the Exam is best viewed through the
+Uniform CPA Examination Blueprints (“Blueprints”) [14], which
+document how concepts and tasks are adapted from Bloom’s
+taxonomy of the cognitive domain [2]. An overview of the
+Exam and sample skills and tasks are provided in Tables 2, 3,
+and 4. The Blueprints are regularly updated by the AICPA and
+are the most detailed, representative outline of the test’s con-
+struction.
+Importantly, many of the tasks detailed in the Blueprints in-
+clude an element of arithmetic. For example, many questions
+that include workpapers or sample financial statements expect
+the test-taker to first determine which numbers to include or ex-
+clude in arithmetic expressions, then to evaluate the resulting
+expression to calculate a specific amount. Sometimes, these
+expressions are as simple as A = L + E, but in many cases,
+they involve more complex expressions based on tables with
+dozens of numbers and related materials. Based on prior re-
+search and experience with LLMs, we strongly suspected that
+GPT-3.5 would struggle with zero-shot quantitative reasoning
+in this context.
+Data
+While there is an active body of research on quantitative
+reasoning with fine-tuning or few-shot contexts [30, 31, 32, 33],
+we constrain our results in this study to zero-shot prompts to
+better assess the “intrinsic” capability of these models. There-
+fore, we prepared two separate assessments to allow us to iso-
+late the arithmetic or quantitative capabilities from other ele-
+ments of the Exam.
+Assessment 1: Sample Exam - Regulation
+The first assessment is intended to approximate the real Uni-
+form CPA Examination using the AICPA’s online, publicly-
+available sample exams.
+These tests “include two multiple-
+choice testlets and three task-based simulation testlets for [...]
+Auditing and Attestation (AUD), Financial Accounting and Re-
+porting (FAR) and Regulation (REG);” the fourth section, BEC,
+is shorter. Between AUD, FAR, and REG, we utilize the REG
+section as it contains the most balanced distribution of skill
+types and quantitative and qualitative reasoning.
+Therefore,
+a test session of the REG exam as provided on the AICPA’s
+site was transcribed on January 3rd, 2023, including correct an-
+swers. All questions are formatted as simple text or, where ev-
+idence or workpapers are formatted in tables or lists, as Mark-
+down.
+This process results in 40 test questions across five testlets.
+Two of these five testlets consist of multiple-choice questions,
+with a total of 15 questions ranging from four to six options
+each. Of the remaining 25 questions, 24 require the test-taker to
+indicate the correct financial amount and one requires the test-
+taker to research authoritative material made available within
+the exam. While we cannot redistribute these test questions di-
+rectly, interested readers can directly access and take the AICPA’s
+online sample exams at no cost.
+A partially-redacted sample question from this assessment
+is provided for reference below:
+Assessment 1: Sample Question
+Question: All taxpayers file their Form 1040 using
+the tax filing status of single. Assume that [...].
+Situation:
+$6,000 - Loss on sale of [...]
+$10,000 - Contribution to the capital [...]
+$3,000 - Write-off of a worthless [...]
+What is the taxpayer’s adjusted gross income?
+Answer: $65,000
+Assessment 2: Synthetic MCQ Assessment
+As noted above, the Uniform CPA Examination is organized
+around Bloom’s cognitive taxonomy [2], which is a widely-
+adopted framework for structuring learning objectives and ca-
+pabilities. The taxonomy is generally conceptualized as a pyra-
+mid divided into six levels: Knowledge, Comprehension, Ap-
+plication, Analysis, Synthesis, and Evaluation or Creation. As
+noted above in Table 2, the AICPA has adapted these skill levels
+into four simpler groups. The top two levels - Evaluation and
+Analysis - not only most frequently feature arithmetic, but in
+practice, are also frequently the most nuanced, contextual tasks
+that real professionals address.
+As an example, tasks like “Evaluate the reasonableness of
+significant accounting estimates [...]” are ones for which, for
+legal and ethical reasons, human oversight will likely remain
+necessary.
+Therefore, we focused this second assessment on the foun-
+dational levels of the AICPA’s skill pyramid - Remembering &
+Understanding and Application. To do so, we reviewed every
+task in the AICPA’s Blueprints, dated October 18, 2021, to iden-
+tify all relevant tasks. For each task, the lead author, a CPA, pre-
+pared at least one question to address each task and skill level
+identified. In sections where there were fewer than 50 relevant
+Blueprint tasks, we randomly sampled tasks and added addi-
+tional questions to ensure that all sections had at least 50 sam-
+ples. While this means that the calculation of overall accuracy
+4
+
+rate overweights sections such as BEC, we are not focused on
+test passage per se in this research and therefore prefer breadth
+and power.
+These questions have been prepared, to the best of our abil-
+ities, to mimic the nature and difficulty of real questions on the
+Exam. In addition to reviewing material provided by the AICPA
+itself, the authors also reviewed material and sample questions
+prepared by McGraw-Hill Education and Becker Professional
+Education to ensure that our test questions were at least as dif-
+ficult and broad as theirs. All questions were drafted solely by
+the authors, and a sample question from each section of this as-
+sessment is provided for reference below.
+Assessment 2: Synthetic REG Question
+Question: Which of the following types of contract
+does not require a written element in order to be
+enforceable?
+A. Contracts for the sale of goods for $500
+or more
+B. Contracts to act as surety
+C. Contracts for the sale of a house
+D. Contracts for leases of land for less than
+one year
+Answer: D
+Assessment 2: Synthetic BEC Question
+Question: Which of the following elements is not
+part of the formula for calculating the cost of
+retained earnings using the Capital Asset Pricing
+Model?
+A. The risk-free rate
+B. The pre-tax cost of long-term debt
+C. The company’s beta coefficient
+D. The market risk premium
+Answer: B
+Assessment 2: Synthetic FAR Question
+Question: Which of the following investment
+types is eligible to be reported in the
+financial statements at amortized cost?
+A. Available-for-sale equity securities
+B. Available-for-sale debt securities
+C. Held-to-maturity debt securities
+D. Trading equity securities
+Answer: C
+Assessment 2: Synthetic AUD Question
+Question: Which of the following disclosures
+related to the fair value of investments in
+securities is required for a nonissuer?
+A. Purchases and issuances for each class of
+investments
+B. Rollfoward of recurring level 3 fair value
+measurements
+C. Disclosures for financial instruments not
+measured at fair value
+D. The range and weighted average of
+significant unobservable inputs
+Answer: A
+These questions, like natural language in the law itself, can
+be subject to pedantic interpretation; for example, in the Au-
+diting and Attestation (AUD) question above, an experienced
+practitioner might qualify choice B by stating that it depends
+on whether it’s a “full rollforward” or a limited number of sep-
+arate elements of the rollforward. Similar to the actual CPA
+Exam, some of our questions may require the selection of the
+“best” option.
+In total, we produced 208 questions across the four sections
+of the Exam. The distribution of these questions is detailed
+in Table 5 below. All questions are available in the online SI
+on GitHub. Like the AICPA’s exam designers themselves, we
+expect that there will be issues with the design or scoring of
+our questions, and we encourage readers to submit additional
+questions or suggested clarifications via corresponding email or
+GitHub. As errata may be detected or new questions accepted,
+updated results may be available in the online SI.
+Assessment
+Section
+Number of Questions
+1
+REG
+40
+2
+AUD
+54
+2
+BEC
+50
+2
+FAR
+51
+2
+REG
+53
+Table 5: Number of AICPA and author-prepared questions per section.
+Methods
+In prior work on the Bar Exam [27], we outlined a method
+for experimentally evaluating OpenAI’s models. For multiple
+choice question (MCQ) assessments in this paper, we follow
+this approach as closely as possible; calculated amounts and
+short answers are compared to the correct answer after stripping
+and reformatting answers. For example, (10, 000), (10000), and
+−10, 000 are identical in the automated scoring of the model’s
+responses.2
+2Parentheses are used as shorthand in the accounting industry for negative
+amounts.
+5
+
+As in prior research, our evaluation is based on generat-
+ing zero-shot prompts for the text-davinci-003 text completion
+API. Unlike in our prior research [27], we are able to fully open-
+source the source code and questions created in Assessment 2.
+While replication of results requires an OpenAI account and ac-
+cepting the AICPA’s terms of use, we have again attempted to
+provide researchers with as much replication detail as is possi-
+ble under the circumstances.
+Prompt Engineering and Responses
+Our ability to understand these large language models is
+constrained both by our limited scientific understanding and
+the proprietary nature of OpenAI’s models [27]. Despite this
+gap, many have documented that such models are unexpectedly
+sensitive to the specific prompts they are provided. The prac-
+tice of writing such prompts is typically referred to as “prompt
+engineering,” and details of prompt engineering are critical to
+replication of studies involving LLMs.
+In this research, we experimented with answer types, con-
+textualization, and justification in prompt engineering [34]. The
+following prompt variations were tested in at least one sam-
+ple, although variations between Assessment 1 and Assessment
+2 are required due to question types. For Assessment 1, the
+prompts define entailment or recall tasks, i.e., where the model
+must select the correct or most correct answer, as well as open-
+ended problems where the model must calculate the correct
+monetary amount. For Assessment 2, all questions are designed
+to evaluate traditional entailment tasks. Complete details are
+available in the source and data in the online SI.
+1. Answer. Ask the model to answer with:
+• its best choice only.
+• its best and worst choices.
+• its top three rank-ordered choices.
+2. Contextualization. Ask the model to imagine it is:
+• taking the CPA exam.
+• designing the CPA exam.
+• an accountant in the United States.
+• a tax professional in the United States.
+• a legal professional in the United States.
+• a Big 4 accountant in the United States.
+3. Justification. Require the model to provide:
+• an explanation of its choices.
+• an explanation and citation to authority or source.
+• an explanation and citation within a specific list of
+authorities or sources.
+Generated prompts are combined with questions and sent to
+the OpenAI API endpoint. The prompt and complete JSON re-
+sponse, including the OpenAI API request ID, are logged for all
+questions for all assessments. The API response is parsed and
+stored for scoring, qualitative analysis, and open source release.
+For scoring, no responses were manually altered or evaluated by
+humans.
+In general, most prompts produced similar performance,
+clustering near the central tendency of 55% noted in Table 8.
+In a number of cases, contextualization or justification resulted
+in models that performed better on one section but worse on an-
+other section. Contextual variations suggest differences in the
+nature of advice between professions. Justification variations
+suggest differences in the complexity or state of codification
+across subject areas. Additional details, complete responses,
+and details regarding phenomena such as hallucination are pro-
+vided in the SI.
+Model (hyper)parameters
+As the AICPA curriculum itself notes, many models are sen-
+sitive to small changes in their inputs, and LLMs are no dif-
+ferent. In addition to prompt sensitivity, they are often highly
+sensitive to the parameters set in training and inference. While
+our ability to intepret results or identify all (hyper)parameters is
+limited by the proprietary nature of GPT, we did evaluate how
+altering some model parameters impacts the performance of the
+model. We do not vary the maximum token output or attempt
+nucleus sampling; however, we do evaluate the following pa-
+rameters for at least one prompt:
+1. temperature: Sampling temperature; 0.0 is deterministic,
+higher is more “random.” We tested values in {0.0, 0.5,
+1.0}.
+2. best of: “Generates [N] completions server-side and re-
+turns the “best” (the one with the highest log probability
+per token).” We tested values in {1, 2, 4}.
+Fine-tuning and Historical Models
+While OpenAI does provide an API for fine-tuning models
+including text-davinci-003, this publication is focused on the
+zero-shot performance of the model itself. Furthermore, based
+on prior experience in similar problems [27], we do not believe
+that fine-tuning text completion at small sample sizes would im-
+prove the models’ performance. In some circumstances, others
+have found success in subsequent supervised or unsupervised
+re-training of some or all layers of an LLM [35][36], while oth-
+ers have documented circumstances in which fine-tuning results
+in unexplained model degradation. In our prior work [27], we
+noted a significant decrease in fine-tuned text-davinci-003 per-
+formance at the scale of our training data. While it is possible
+that this performance decrease is explained by the 50% head
+layer contraction required by OpenAI’s API, we are unable to
+test further without access to details of fine-tuning or resulting
+weights.
+In addition to text-davinci-003, OpenAI also makes a num-
+ber of other models available through its API, including smaller
+and older iterations of the GPT family. We repeated our testing
+with the text-davinci-001, text-curie-001, text-babbage-001,
+and text-ada-001 models provided through the OpenAI API.
+6
+
+Results
+In total, across all prompts and parameters tested, we asked
+text-davinci-003 to answer over 50,000 questions in more than
+700 independent assessment sessions. Details of the number
+of sessions and parameter values tested are described below in
+each assessment and in the online SI. The range of performance
+values observed over all experiments is summarized in Table 6.
+Correct Rates by Question Type and Assessment
+Assessment
+Amount
+MCQ
+Short Answer
+Assessment 1
+5.7 - 9.4%
+22.3 - 28.1%
+0%
+Assessment 2
+N/A
+50.0 - 57.6%
+N/A
+Table 6: Correct rates by question type and assessment as measured by all-
+experiment range of mean prompt performance between Assessment 1 and As-
+sessment 2. Baseline for Multiple Choice is 22.67% for Assessment 1, 25% for
+Assessment 2. Description of best prompts and parameters is provided below
+and prompt details are available in SI.
+Assessment 1
+As expected, the quantitative reasoning and arithmetic re-
+quired in Assessment 1 resulted in substantially lower zero-shot
+performance than observed in Assessment 2. Out of 24 ques-
+tions that required the test-taker to provide a numeric answer
+based on facts and work papers, GPT-3.5 frequently only an-
+swered one, two, or three questions correctly, resulting in an
+average range across all parameters and prompts of 5.7 to 9.4%.
+While it is arguable whether 0% is the true baseline for this task,
+it is clear that such zero-shot performance is not on par with hu-
+man test-takers.
+GPT-3.5 also struggled with arithmetic on the 15 MCQs on
+Assessment 1, scoring above random chance for some, but not
+all, prompts and parameters. As a number of questions include
+more than four choices, the true baseline rate of guessing is
+22.67%, not 25%, but despite this, the best prompts and param-
+eters were only 4-6% above the baseline rate.
+Based on a qualitative review of these questions and the
+model’s responses, we believe that performance could be im-
+proved somewhat in few-shot evaluations. Further, we believe
+that even some zero-shot performance improvements could be
+achieved by expanding the prompt to include “scratchpads” for
+common relationships or equations [37], as might be seen on
+problems that feature common workpapers like a statement of
+cash flows; however, in this paper, we focus on a zero-shot,
+“out-of-the-box” evaluation, and so these improvements are left
+for future research.
+Assessment 2
+As discussed in Assessment 2, we created 208 MCQs for
+Assessment 2 to evaluate GPT-3.5’s capabilities at the founda-
+tion of knowledge work. Each of these 208 questions has four
+options, and therefore, the baseline guessing rate for the model
+is exactly 25%. We assessed GPT-3.5 on 208-question assess-
+ment exactly 180 times - three samples for each combination of
+10 prompts, three temperature (T) values, and two best of (n)
+parameter values (3 · 10 · 3 · 2). Across these 10 prompts, mean
+performance ranged between 51.1% and 56.9%, with a worst
+run of 50.0% (Prompt 13, T = 1.0) and a best run of 57.6%
+(Prompt 16, T = 0.0). We did not find significant differences
+between n parameter values in this assessment.
+Section
+Accuracy
+Accuracy - Top Two
+AUD
+57.1%
+84.9%
+BEC
+69.7%
+85.7%
+FAR
+51.0%
+82.4%
+REG
+53.1%
+75.8%
+Table 7: Accuracy of GPT-3.5 by section of AICPA Exam Blueprints for best
+prompt and parameter, with correct rate including second-best answer in paren-
+theses. Passage rates are provided in Table 1 below for reference, but should
+not be directly compared with model accuracy rates for the reasons discussed
+above.
+Table 7, Table 1, and Figure 1 show the performance of this
+best prompt and parameter value, including the average per-
+centage of correct questions by section and the average pas-
+sage rate for test-takers in 2022 as reported by [29]. Over-
+all, GPT-3.5 is demonstrating performance significantly in ex-
+cess of guessing, achieving approximately 70% in questions on
+Business Environment and Concepts (BEC), 57% for Auditing
+and Attestation (AUD), 53% for Regulation (REG), and 51%
+for Financial Accounting and Reporting (FAR). Furthermore,
+as seen in prior research [27], GPT-3.5 demonstrates strong
+non-entailment performance as represented by its rank order-
+ing of choices. The model’s top two answers are correct over
+82% of the time, significantly in excess of the 50% baseline.
+While we did not qualitatively code all 208 question for the
+applicable AICPA skill level, we did review all 53 questions
+from the Regulation section in Assessment 2. We found that
+at least 23 of the 53 questions (≈43%) require some degree of
+Application or Analysis. While these skill levels may be sub-
+jective in the context of realistic questions, we encourage read-
+ers to examine the complete set of 208 questions in the SI for
+themselves and to self-assess their own performance to set ex-
+pectations regarding task type and difficulty.
+We do not have a head-to-head comparison between real
+test-takers and GPT-3.5 for Assessment 2. Based on our ex-
+perience, however, we believe that these questions are at least
+as difficult as the real Remembering & Understanding and Ap-
+plication questions on the Exam. Further, the tasks tested in
+Assessment 2 also account for the vast majority of tasks and
+types of tasks covered in the AICPA Blueprints. In addition
+to reviewing models for single correct answers, some prompts
+also required models to provide explanations or justifications.
+We performed a qualitative review of explanations and justifi-
+cations for a sample of sessions, and found that more than half
+of the model’s correct answers were also correctly explained
+with the correct reference or authority. Interested readers are di-
+rected to the online SI for thousands of examples of responses
+from the model. Out of all explanations, including incorrect
+ones, explanations included at least one hallucinated reference
+or authority in approximately 37% of the time. Research is on-
+going on the optimal degree of hallucination and techniques for
+mitigating unwanted hallucination [38], and we will continue
+to explore these questions and applications in future work.
+7
+
+AUD
+BEC
+FAR
+REG
+Section
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+Correct Rate
+Random Chance
+GPT-3.5 Average
+GPT-3.5 Performance on Assessment 2 by Section
+GPT Top Two Choices
+GPT First Choice
+Figure 1: Performance of GPT-3.5 by section of AICPA Exam Blueprints for best prompt and parameter, with correct rate including second-best answer in dashed
+region. Error bars are ±1 standard error of the mean. Note that GPT-3.5 is not assessed on Analysis or Evaluation tasks, unlike human test-takers, and that the
+percentage of questions correct does not scale linearly with score or passage.
+GPT-2
+ada-001
+curie-001
+babbage-001
+davinci-001
+davinci-003
+Model
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+Correct Rate
+Random Chance
+Q1 2019
+Q4 2022
+Progression of GPT Models on Assessment 2 (CPA Exam)
+Figure 2: Comparison of model performance across GPT-3 generations. For text-davinci-003, the average is reported across all runs; for other models, a subset of
+representative prompts and parameters were included. GPT-2 was unable to reliably respond to the prompt as instructed and questions were larger than its maximum
+input token length. More details are available in source and data in the online SI.
+8
+
+Model
+Correct
+text-davinci-003
+55.1
+text-davinci-001
+29.9
+text-babbage-001
+25.2
+text-curie-001
+20.4
+text-ada-001
+9.7
+Table 8: Comparison of model performance across GPT-3 generations. For
+text-davinci-003, the average is reported across all runs; for other models, a
+subset of representative prompts and parameters were included. More details
+are available in source and data in the online SI.
+GPT Model Progression
+In prior work [27], we noted that text-davinci-003 demon-
+strated material improvements from prior generations of GPT
+models. In this work, we also compare our results against older
+or smaller GPT-3 models. Table 8 and Figure 2 summarize
+these findings, demonstrating a qualitatively-identical story from
+our work on the Bar Exam. Only text-davinci-001 exhibits the
+ability to follow instructions and answer above random chance,
+and between 001 and 003, the spread over random guessing has
+increased from less than 5% to over 30%.
+Conclusion and Future Work
+In this paper, we document and develop two assessments
+of knowledge worker readiness based on the AICPA’s Uniform
+CPA Examination Blueprints. Assessment 1 is a sample Regu-
+lation test as provided by the AICPA, including quantitative rea-
+soning and calculations; Assessment 2 covers foundational skill
+levels, excluding quantitative reasoning and calculations, for all
+four sections of the Blueprints. In total, these assessments cover
+a broad, practical curriculum including law, finance, account-
+ing, and technology. We then experimentally evaluate GPT-3.5
+on these two assessments, including detailed steps to replicate
+this evaluation, and share source code and data for all questions
+not covered by copyright.
+First, we find that text-davinci-003 achieves a correct rate
+of 14.4% on Assessment, significantly underperforming test-
+takers. As many authors have documented in research on large
+language models [31, 32, 33], arithmetic and quantitative rea-
+soning are often outside the scope of zero-shot use cases, and
+these results are consistent with these prior findings.
+As arithmetic and quantitative reasoning are the subjects of
+substantial active research, we look forward to exploring zero-
+shot approaches as new models or techniques become available.
+Further, as many industrial applications will support iterative or
+few-shot approaches, we are continuing to investigate applied
+use cases like the calculation of financial or operational met-
+rics or the analysis of specific financial statements using more
+mature techniques like [39].
+Second, we find that text-davinci-003 can achieve an accu-
+racy of 57% on Assessment 2, significantly better than a 25%
+guessing rate, and approaching or on par with anecdotal test-
+taker performance. It also demonstrates strong non-entailment
+capabilities and improving explanation capabilities, as its top
+two answers are correct 82% of the time and explanations are
+correct more often than not. While this assessment is not iden-
+tical to the CPA Exam and the AICPA does not publish directly
+comparable statistics, approximately 45-55% of test-takers fail
+the exams annually, as an indication of general difficulty. All
+questions in this assessment are available for readers to review
+and self-assess, and we encourage others to suggest improve-
+ments or perform their own assessment on this material.
+Finally, as in prior research, we find that recent generations
+of GPT-3 demonstrate material improvements on this assess-
+ment. While text-ada-001 could barely follow instructions and
+text-davinci-001 only exceeded random chance by 5%, text-
+davinci-003 is now approaching human performance on this as-
+sessment.
+As organizations and institutions around the world depend
+on knowledge workers to navigate an increasingly complex le-
+gal and financial landscape [40, 41], it is critical that we de-
+velop tools that can help safely, effectively meet this demand
+for knowledge work. Our findings strongly suggest that future
+large language models have the potential to transform the qual-
+ity and efficiency of knowledge work at least as much as search
+engines did at the turn of the 21st century.
+Acknowledgments
+Although the original draft of this paper was written by the
+authors, portions of this paper were fine-tuned by text-davinci-
+003 for formatting and clarity.
+Supplementary Information
+Almost all of the material used in the creation and presen-
+tation of this research is available in the online Supplementary
+Information (SI) at the following URL:
+https://github.com/mjbommar/gpt-as-knowledge-worker.
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+

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+Semiparametric Regression for Spatial Data via
+Deep Learning
+Kexuan Li ∗, Jun Zhu †, Anthony R. Ives ‡,
+Volker C. Radeloff §, and Fangfang Wang¶
+January 11, 2023
+Abstract
+In this work, we propose a deep learning-based method to perform semiparametric re-
+gression analysis for spatially dependent data. To be specific, we use a sparsely connected
+deep neural network with rectified linear unit (ReLU) activation function to estimate the
+unknown regression function that describes the relationship between response and covariates
+in the presence of spatial dependence. Under some mild conditions, the estimator is proven
+to be consistent, and the rate of convergence is determined by three factors: (1) the archi-
+tecture of neural network class, (2) the smoothness and (intrinsic) dimension of true mean
+function, and (3) the magnitude of spatial dependence. Our method can handle well large
+data set owing to the stochastic gradient descent optimization algorithm. Simulation studies
+on synthetic data are conducted to assess the finite sample performance, the results of which
+indicate that the proposed method is capable of picking up the intricate relationship between
+response and covariates. Finally, a real data analysis is provided to demonstrate the validity
+and effectiveness of the proposed method.
+Keywords: Semiparametric regression; Spatially dependent data; Deep Neural Networks; Stochas-
+tic gradient descent.
+∗[email protected], Global Analytics and Data Sciences, Biogen, Cambridge, Massachusetts, US.
+†[email protected], Department of Statistics, University of Wisconsin-Madison.
+‡[email protected], Department of Integrative Biology, University of Wisconsin-Madison.
+§radeloff@wisc.edu, Department of Forest and Wildlife Ecology, University of Wisconsin-Madison.
+¶[email protected], Department of Mathematical Sciences, Worcester Polytechnic Institute.
+1
+arXiv:2301.03747v1  [stat.ML]  10 Jan 2023
+
+1
+Introduction
+With recent advances in remote sensing technology and geographical sciences, there has been
+a considerable interest in modeling spatially referenced data. The purpose of this paper is to
+develop new methodology that captures complex structures in such data via deep neural networks
+and Gaussian random fields. In addition, we provide a theoretical understanding of deep neural
+networks for spatially dependent data.
+In recent years, deep neural network (DNN) has made a great breakthrough in many fields,
+such as computer vision (He et al., 2016), dynamics system (Li et al., 2021), natural language
+processing (Bahdanau et al., 2014), drug discovery and toxicology (Jim´enez-Luna et al., 2020),
+and variable selection (Li et al., 2022; Li, 2022). Besides its successful applications, there has also
+been great progress on theoretical development of deep learning. Liu et al. (2020) and Schmidt-
+Hieber (2020) proved that the neural network estimator achieves the optimal (up to a logarithmic
+factor) minimax rate of convergence. Liu et al. (2022) further removed the logarithmic term and
+achieved the exact optimal nonparametric convergence rate. One of the appealing features of deep
+neural network is that it can circumvent the curse of dimensionality under some mild conditions.
+Owing to the superior performance and theoretical guarantees of deep learning, applying deep
+learning to spatial data has also drawn much attention.
+For example, Zammit-Mangion and
+Wikle (2020) fitted the integro-difference equation through convolutional neural networks and
+obtained probabilistic spatio-temporal forecasting. Zammit-Mangion et al. (2021) constructed a
+deep probabilistic architecture to model nonstationary spatial processes using warping approach.
+Mateu and Jalilian (2022) used variational autoencoder generative neural networks to analyze
+spatio-temporal point processes. Kirkwood et al. (2022) applied Bayesian deep neural network to
+spatial interpolation. However, there is a lack of theoretical understanding of the aforementioned
+work, which we will address in this paper.
+In addition, we model spatial dependence by Gaussian random fields and develop model esti-
+mation with computational efficiency. Due to technological advances in data collecting process, the
+size of spatial datasets are massive and traditional statistical methods encounter two challenges.
+One challenge is the aggravated computational burden. To reduce computation cost, various meth-
+ods have been developed, such as covariance tapering, fixed rank kriging, and Gaussian Markov
+2
+
+random fields (see Sun et al. (2012) for a comprehensive review). The other challenge is data
+storage and data collection. Many spatial datasets are not only big, but are also generated by
+different sources or in an online fashion such that the observations are generated one-by-one. In
+both cases, we cannot process entire datasets at once. To overcome these two challenges, Robbins
+and Monro (1951) proposed a computationally scalable algorithm called stochastic gradient de-
+scent (SGD) and achieved great success in many areas. Instead of evaluating the actual gradient
+based on an entire dataset, SGD estimates the gradient using only one observation which makes
+it computationally feasible with large scale data and streaming data. Its statistical inferential
+properties have also been studied by many researchers (Su and Zhu, 2018; Liu et al., 2021).
+To meet these challenges, here we develop deep learning-based semiparametric regression for
+spatial data. Specifically, we use a sparsely connected feedforward neural network to fit the regres-
+sion model, where the spatial dependence is captured by Gaussian random fields. By assuming a
+compositional structure on the regression function, the consistency of the neural network estima-
+tor is guaranteed. The advantages of the proposed method are fourfold. First, we do not assume
+any parametric functional form for the regression function, allowing the true mean function to
+be nonlinear or with complex interactions. This is an improvement over many of the existing
+parametric, semiparametric, or nonparametric approaches (Hastie and Tibshirani, 1993; Fan and
+Zhang, 1999; Mu et al., 2018; Kim and Wang, 2021; Lee, 2002; Robinson, 2011; Jenish, 2012; Li,
+2016; Lu and Tjøstheim, 2014; Kurisu, 2019, 2022). Second, under some mild technical conditions,
+we show that the estimator is consistent. To the best of our knowledge, this is the first theoretical
+result in deep neural network for spatially dependent data. Third, the convergence rate is free of
+the input dimension, which means our estimator does not suffer from the curse of dimensionality.
+Finally, owing to the appealing properties of SGD, our method is feasible for large scale dataset
+and streaming data.
+The remainder of the paper is organized as follows. Section 2 formulates the statistical problem
+and presents the deep neural network estimator.
+The computational aspects and theoretical
+properties of the estimator are given in Section 3. Section 4 evaluates the finite-sample performance
+of the proposed estimator by a simulation study. We apply our method to a real-world dataset in
+Section 5. Technical details are provided in Appendix.
+3
+
+2
+Model and Estimator
+In this section, we first formulate the problem and then present the proposed estimator under a
+deep learning framework.
+2.1
+Model Setup
+For a spatial domain of interest S, we consider the following semiparametric spatial regression
+model:
+y(s) = f0(x(s)) + e1(s) + e2(s), s ∈ S
+(1)
+where f0 : [0, 1]d → R is an unknown mean function of interest, x(s) = (x1(s), . . . , xd(s))⊤
+represents a d-dimensional vector of covariates at location s with xi(s) ∈ [0, 1], e1(s) is a mean zero
+Gaussian random field with covariance function γ(s, s′), s, s′ ∈ S, and e2(s) is a spatial Gaussian
+white noise process with mean 0 and variance σ2. Furthermore, we assume that e1(s), e2(s),
+and x(s) are independent of each other. Thus the observation y(s) comprises three components:
+large-scale trend f0(x(s)), small-scale spatial variation e1(s), and measurement error e2(s); see,
+for instance, Cressie and Johannesson (2008).
+In the spatial statistics literature, it is popular to focus on predicting the hidden spatial pro-
+cess y(s)∗ = f0(x(s)) + e1(s) using the observed information (Cressie and Johannesson, 2008).
+However, the primary interest of this paper is to estimate the large-scale trend f0(x(s)), where
+the relationship between the hidden spatial process and the covariates could be complex in na-
+ture.
+To capture such a complex relationship, we assume that f0 is a composition of several
+functions inspired by neural networks characteristics (Schmidt-Hieber, 2020). H¨older smoothness
+(see Definition 1 in Appendix) is a commonly used smoothness assumption for regression func-
+tion in nonparametric and semiparametric literature. Thus it is natural to assume the true mean
+function f0 is a composition of H¨older smooth functions, which is formally stated in the following
+assumption.
+Assumption 1. The function f0 : Rd → R has a compositional structure with parameters
+(L∗, r, ˜r, β, a, b, C) where L∗ ∈ Z+, r = (r0, . . . , rL∗+1)⊤ ∈ ZL∗+2
++
+with r0 = d and rL∗+1 = 1, ˜r =
+(˜r0, . . . , ˜rL∗)⊤ ∈ ZL∗+1
++
+, β = (β0, . . . , βL∗)⊤ ∈ RL∗+1
++
+, a = (a0, . . . , aL∗+1)⊤, b = (b0, . . . , bL∗+1)⊤ ∈
+4
+
+RL∗+2, and C = (C0, . . . , CL∗)⊤ ∈ RL∗+1
++
+; that is,
+f0(z) = gL∗ ◦ . . . ◦ g1 ◦ g0(z),
+for z ∈ [a0, b0]r0
+where Z+, R+ denote the sets of positive integers and positive real numbers, respectively, gi =
+(gi,1, . . . , gi,ri+1)⊤ : [ai, bi]ri → [ai+1, bi+1]ri+1 for some |ai|, |bi| ≤ Ci and the functions gi,j :
+[ai, bi]˜ri → [ai+1, bi+1] are (βi, Ci)-H¨older smooth only relying on ˜ri variables and ˜ri ≤ ri.
+Without loss of generality, we assume Ci > 1 in Assumption 1. The parameter L∗ refers to
+the total number of layers, i.e., the number of composite functions, r is the whole number of
+variables in each layer, whereas ˜r is the number of “active” variables in each layer. The two
+parameter vectors β and C pertain to the H¨older smoothness in each layer, while a and b define
+the domain of gi in the ith layer. For the rest of the paper, we will use CS(L∗, r, ˜r, β, a, b, C)
+to denote the class of functions that have a compositional structure as specified in Assumption 1
+with parameters (L∗, r, ˜r, β, a, b, C).
+It is worth mentioning that Assumption 1 is commonly adopted in deep learning literature; see
+Bauer and Kohler (2019), Schmidt-Hieber (2020), Kohler and Langer (2021), Liu et al. (2020), Li
+et al. (2021), among others. This compositional structure covers a wide range of function classes
+including the generalized additive model.
+Example 1. The generalized additive model is a generalized linear model with a linear predictor
+involving a sum of smooth functions of covariates (Hastie and Tibshirani, 1986). Suppose f(z) =
+ϕ(�d
+i=1 hi(zi)), where ϕ(·) is (βϕ, Cϕ)-H¨older smooth and hi(·) are (βh, Ch)-H¨older smooth, for
+some (βϕ, Cϕ) and (βh, Ch).
+Clearly, f(z) can be written as a composition of three functions
+f(z) = g2 ◦ g1 ◦ g0(z) with g0(z1, . . . , zd) = (h1(z1), . . . , hd(zd))⊤, g1(z1, . . . , zd) = �d
+i=1 zi, and
+g2(z) = ϕ(z). Here, L∗ = 2, r = (d, d, 1, 1)⊤, ˜r = (d, d, 1)⊤, and β = (βh, ∞, βϕ)⊤.
+2.2
+Deep Neural Network (DNN) Estimator
+In this paper, we consider estimating the unknown function f0 via a deep neural network owing to
+the complexity of f0 and the flexibility of neural networks. So before presenting our main results,
+we first briefly review the neural network terminologies pertaining to this work.
+5
+
+An activation function is a nonlinear function used to learn the complex pattern from data. In
+this paper, we focus on the Rectified Linear Unit (ReLU) shifted activation function which is de-
+fined as σv(z) = (σ(z1 −v1), . . . , σ(zd −vd))⊤, where σ(s) = max{0, s} and z = (z1, . . . , zd)⊤ ∈ Rd.
+ReLU activation function enjoys both theoretical and computational advantages. The projection
+property σ ◦σ = σ can facilitate the proof of consistency, while ReLU activation function can help
+avoid vanishing gradient problem. The ReLU feedforward neural network f(z, W, v) is given by
+f(z, W, v) = WLσvL . . . W1σv1W0z,
+z ∈ Rp0,
+(2)
+where {(W0, . . . , WL) : Wl ∈ Rpl+1×pl, 0 ≤ l ≤ L} is the collection of weight matrices, {(v1, . . . , vL) :
+vl ∈ Rpl, 1 ≤ l ≤ L} is the collection of so-called biases (in the neural network literature), and
+σvl(·), 1 ≤ l ≤ L, are the ReLU shifted activation functions. Here, L measures the number of
+hidden layers, i.e., the length of the network, while pj is the number of units in each layer, i.e., the
+depth of the network. When using a ReLU feedforward neural network to estimate the regression
+problem (1), we need to have p0 = d and pL+1 = 1; and the parameters that need to be estimated
+are the weight matrices (Wj)j=0,...,L and the biases (vj)j=1,...,L.
+By definition, a ReLU feedforward neural network can be written as a composition of simple
+nonlinear functions; that is,
+f(z, W, v) = gL ◦ . . . ◦ g1 ◦ g0(z),
+z ∈ Rp0,
+where gi(z) = Wiσvi(z), z ∈ Rpi, i = 1, . . . , L, and g0(z) = W0z, z ∈ Rp0. Unlike traditional
+function approximation theory where a complex function is considered as an infinite sum of sim-
+pler functions (such as Tayler series, Fourier Series, Chebyshev approximation, etc.), deep neural
+networks approximate a complex function via compositions, i.e., approximating the function by
+compositing simpler functions (Lu et al., 2020; Farrell et al., 2021; Yarotsky, 2017).
+Thus, a
+composite function can be well approximated by a feedforward neural network. That is why we
+assume the true mean function f0 has a compositional structure.
+In practice, the length and depth of the networks can be extremely large, thereby easily causing
+overfitting. To overcome this problem, a common practice in deep learning is to randomly set some
+neurons to zero, which is called dropout. Therefore, it is natural to assume the network space is
+sparse and all the parameters are bounded by one, where the latter can be achieved by dividing
+6
+
+all the weights by the maximum weight (Bauer and Kohler, 2019; Schmidt-Hieber, 2020; Kohler
+and Langer, 2021). As such, we consider the following sparse neural network class with bounded
+weights
+F(L, p, τ, F) =
+�
+f is of form (2) : max
+j=0,...,L ∥Wj∥∞ + |vj|∞ ≤ 1,
+L
+�
+j=0
+(∥Wj∥0 + |vj|0) ≤ τ, ∥f∥∞ ≤ F
+�
+,
+(3)
+where p = (p0, . . . , pL+1) with p0 = d and pL+1 = 1, and v0 is a vector of zeros. This class of
+neural networks is also adopted in Schmidt-Hieber (2020), Liu et al. (2020), and Li et al. (2021).
+Suppose that the process y(·) is observed at a finite number of spatial locations {s1, . . . , sn}
+in S. The desired DNN estimator of f0 in Model (1) is a sparse neural network in F(L, p, τ, F)
+with the smallest empirical risk; that is,
+�fglobal(�
+W, �v) =
+argmin
+f∈F(L,p,τ,F)
+n−1
+n
+�
+i=1
+(y(si) − f(x(si))2.
+(4)
+For simplicity, we sometimes write �fglobal for �fglobal(�
+W, �v) if no confusion arises.
+3
+Computation and Theoretical Results
+In this section, we describe the computational procedure used to optimize the objective function
+(4) and present the main theoretical results.
+3.1
+Computational Aspects
+Because (4) does not have an exact solution, we use a stochastic gradient descent (SGD)-based
+algorithm to optimize (4). In contrast to a gradient descent algorithm which requires a full dataset
+to estimate gradients in each iteration, SGD or mini-batch gradient descent only needs an access
+to a subset of observations during each update, which is capable of training relatively complex
+models for large datasets and computationally feasible with streaming data. Albeit successful
+applications in machine learning and deep learning, SGD still suffers from some potential problems.
+For example, the rate of convergence to the minima is slow; the performance is very sensitive to
+tuning parameters. To circumvent these problems, various methods have been proposed, such as
+RMSprop and Adam (Kingma and Ba, 2014). In this paper, we use Adam optimizer to solve (4).
+7
+
+During the training process, there are many hyper-parameters to tune in our approach: the
+number of layers L, the number of neurons in each layer p, the sparse parameter τ, and the
+learning rate. These hyper-parameters play an important role in the learning process. However,
+it is challenging to determine the values of hyper-parameters without domain knowledge.
+In
+particular, it is challenging to control the sparse parameter τ directly in the training process. Thus,
+we add an ℓ1-regularization penalty to control the number of inactive neurons in the network. The
+idea of adding a sparse regularization to hidden layers in deep learning is very common; see, for
+instance, Scardapane et al. (2017) and Lemhadri et al. (2021). In this paper, we use a 5-fold
+cross-validation to select tuning parameters.
+3.2
+Theoretical Results
+Recall that the minimizer of (4), �fglobal, is practically unattainable and we use an SGD-based
+algorithm to minimize the objective function (4), which may converge to a local minimum. The
+actual estimator obtained by minimizing (4) is denoted by �flocal ∈ F(L, p, τ, F). We define the
+difference between the expected empirical risks of �fglobal and �flocal as
+∆n( �flocal) .= Ef0
+�
+1
+n
+n
+�
+i=1
+(y(si) − �flocal(x(si))2 −
+inf
+˜f∈F(L,p,τ,F)
+1
+n
+n
+�
+i=1
+(y(si) − ˜f(x(si))2
+�
+= Ef0
+�
+1
+n
+n
+�
+i=1
+(y(si) − �flocal(x(si))2 − 1
+n
+n
+�
+i=1
+(y(si) − �fglobal(x(si))2
+�
+,
+(5)
+where Ef0 stands for the expectation with respect to the true regression function f0. For any
+�f ∈ F(L, p, τ, F), we consider the following estimation error:
+Rn( �f, f0) .= Ef0
+�
+1
+n
+n
+�
+i=1
+� �f(x(si)) − f0(x(si))
+�2
+�
+.
+(6)
+The oracle-type theorem below gives an upper bound for the estimation error.
+Theorem 1. Suppose that the unknown true mean function f0 in (1) satisfies ∥f0∥∞ ≤ F for
+some F ≥ 1. For any δ, ϵ ∈ (0, 1] and �f ∈ F(L, p, τ, F), the following oracle inequality holds:
+Rn( �f, f0) ≲(1 + ε)
+�
+inf
+˜f∈F(L,p,τ,F) ∥ ˜f − f0∥2
+∞ + ζn,ε,δ + ∆n( �f)
+�
+,
+8
+
+where
+ζn,ε,δ ≍1
+ε
+�
+δ
+�
+n−1 tr(Γn) + 2
+�
+n−1 tr(Γ2
+n) + 3σ
+�
++ τ
+n (log(L/δ) + L log τ) (n−1 tr(Γ2
+n) + σ2 + 1)
+�
+,
+and Γn = [γ(si, si′)]1≤i,i′≤n.
+The convergence rate in Theorem 1 is determined by three components. The first component
+inf ˜f∈F(L,p,τ,F) ∥ ˜f −f0∥2
+∞ measures the distance between the neural network class F(L, p, τ, F) and
+f0, i.e., the approximation error. The second term ζn,ε,δ pertains to the estimation error, and
+∆n( �f) is owing to the difference between �f and the oracle neural network estimator �fglobal. It is
+worth noting that the upper bound in Theorem 1 does not depend on the network architecture
+parameter p, i.e., the width of the network, in that the network is sparse and its “actual” width
+is controlled by the sparsity parameter τ. To see this, after removing all the inactive neurons, it
+is straightforward to show that F(L, p, τ, F) = F(L, (p0, p1 ∧ τ, . . . , pL ∧ τ, pL+1), τ, F) (Schmidt-
+Hieber, 2020).
+Next, we turn to the consistency of the DNN estimator �flocal for f0 ∈ CS(L∗, r, ˜r, β, a, b, C).
+In nonparametric regression, the estimation convergence rate is heavily affected by the smooth-
+ness of the function.
+Consider the class of composite functions CS(L∗, r, ˜r, β, a, b, C).
+Let
+β∗
+i = βi
+�L∗
+s=i+1(βs ∧ 1) for i = 0, . . . , L∗ and i∗ = argmin0≤i≤L∗ β∗
+i /˜ri, with the convention
+�L∗
+s=L∗+1(βs ∧ 1) = 1. Then β∗ = β∗
+i∗ and r∗ = ˜ri∗ are known as the intrinsic smoothness and
+intrinsic dimension of f ∈ CS(L∗, r, ˜r, β, a, b, C). These quantities play an important role in
+controlling the convergence rate of the estimator. To better understand β∗
+i and i∗, think about the
+composite function from the ith to the last layer, i.e., hi(z) = gL∗ ◦. . .◦gi+1 ◦gi(z) : [ai, bi]ri → R;
+then β∗
+i can be viewed as the smoothness of hi and i∗ is the layer of the least smoothness after
+rescaled by the respective number of “active” variables ˜ri, i = 0, . . . , L∗. The following theorem
+establishes the consistency of �flocal as an estimator of f0 and its convergence rate in the presence
+of spatial dependence.
+Theorem 2. Suppose Assumption 1 is satisfied, i.e., f0 ∈ CS(L∗, r, ˜r, β, a, b, C). Let �flocal ∈
+F(L, p, τ, F) be an estimator of f0. Further assume that F ≥ maxi=0,...,L∗(Ci, 1), N .= mini=1,...,L pi ≥
+6η maxi=0,...,L∗(βi + 1)˜ri ∨ ( ˜Ci + 1)e˜ri where η = maxi=0,...,L∗(ri+1(˜ri + ⌈βi⌉)), and τ ≲ LN. Then
+we have
+Rn( �flocal, f0) ≲ ςn,
+9
+
+where
+ςn ≍ (N2−L)2 �L∗
+l=1 βl∧1 + N − 2β∗
+r∗ + (tr(Γ2
+n) + n)(LN log(Ln2) + L2N log(LN))
+n2
++ ∆n( �flocal),
+and ˜Ci are constants only depending on C, a, b, and Γn = [γ(si, si′)]1≤i,i′≤n.
+The consistency of �flocal can be achieved by, for instance, letting L ≍ log(n), N ≍ n
+r∗
+2β∗+r∗ , tr(Γ2
+n) =
+o(n
+4β∗+r∗
+2β∗+r∗ (log n)−3), and ∆n( �flocal) = o(1), as a result of which ςn ≍ n−
+2β∗
+2β∗+r∗ (log n)3 + ∆n( �flocal) =
+o(1). As expected, the rate of convergence is affected by the intrinsic smoothness and intrinsic
+dimension of CS(L∗, r, ˜r, β, a, b, C), the architecture of the neural network F(L, p, τ, F), and the
+magnitude of the spatial dependence.
+4
+Simulation Study
+In this section, we evaluate the finite sample performance of the proposed DNN estimator through
+a set of simulation studies. Two different simulation designs are considered, and for each design,
+we generate 100 independent data sets.
+In the first design, the spatial domain of interest S is in R. To be specific, we generate data
+from the following model
+y(si) = f0(x(si)) + e1(si) + e2(si), si ∈ [0, D], i = 1, 2, . . . , n,
+and
+x(si) = (x1(si), . . . , x5(si))⊤ =
+�
+si/D, sin(10si/D), (si/D)2, exp(3si/D), (si/D + 1)−1�⊤ ∈ R5,
+with the true mean function f0(x(si)) = �5
+j=1 xj(si). The small-scale spatial variation e1(·) is
+a zero-mean stationary and isotropic Gaussian process with an exponential covariance function
+γ(si, sj) = exp(−|si − sj|/ρ) and the range parameter ρ = 0.1, 0.5, 1. The measurement error
+e2(·) is standard normal distributed and independent of e1(·). It is worth mentioning that the
+covariates are location dependent.
+We consider two different spatial domains: fixed domain and expanding domain. For the fixed
+domain, S = [0, 1] is a fixed interval, i.e., D = 1, whereas for the expanding domain, the spatial
+10
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+5
+10
+15
+20
+(a) Fixed domain: S = [0, 1].
+0
+2
+4
+6
+8
+10
+0
+5
+10
+15
+20
+s
+f0
+(b) Expanding domain: S = [0, 10].
+Figure 1: The estimated mean function and 95% pointwise simulation intervals using our method
+in Simulation Design 1 with n = 100, ρ = 0.5. In both plots, the solid line is the true mean
+function, and the two dashed lines are the 95% pointwise simulation intervals. The grey lines are
+the estimated mean functions from each replication.
+domain S = [0, D] increases with the sample size n. The n observations are equally spaced over
+the region. In both scenarios, we let n = 100, 200, 300, and accordingly, D = 10, 20, 30 in the
+expanding domain case.
+In the second design, the mean function is defined on R2, given by
+f0(x(si)) =β1x1(si)x2(si) + β2x2(si)2 sin(x3(si)) + β3 exp(x4(si)) max(x5(si), 0)
++
+β4
+sign x4(si)(10 + x5(si)) + β5 tanh(x1(si)),
+si ∈ [0, D]2, i = 1, . . . , n,
+(7)
+where the coefficients βj, j = 1, . . . , 5, are drawn from U(1, 2). The covariates at each location
+are generated from standard normal distributions with a cross-covariate correlation of 0.5 and
+the covariates at different locations are assumed to be independent. We further normalize each
+covariate to have zero mean and unit variance.
+The mean function f0 is nonlinear, featuring
+interactions among the covariates. We simulate y(si) according to (1) with e1(si) and e2(si) similar
+to those in Design 1. That is, e1(si) is a zero-mean stationary and isotropic Gaussian process on
+R2 with an exponential covariance function γ(si, sj) = exp(−|si − sj|/ρ) and ρ = 0.1, 0.5, 1, and
+11
+
+e2(si) ∼ N(0, 1).
+Similar to the first design, we consider two types of spatial domain: fixed domain, i.e., D = 1
+and expanding domain, i.e., D = 10, 20, 30. In both cases, we have n = 100, 400, 900, and all the
+locations are equally spaced over [0, D]2.
+4.1
+Estimating f0 via other methods
+We also compare the proposed DNN estimator with three state-of-the-art estimators in the litera-
+ture. The first estimator of f0 is based on the Gaussian process-based spatially varying coefficient
+model (GP-SVC) which is given by
+y(s) = β1(s)x1(s) + . . . , +βp(s)xp(s) + ϵ, ϵ ∼ N(0, τ 2), s ∈ S,
+and the spatially varying coefficient βj(·) is the sum of a fixed effect and a random effect. That
+is, βj(s) = µj + ηj(s), where µj is a non-random fixed effect and ηj(·) is a zero-mean Gaussian
+process with an isotropic covariance function c(·; θj). In this work, we use the popular Mat´ern
+covariance function defined as
+c
+�
+r; ρ, ν, σ2�
+= σ2 21−ν
+Γ(ν)
+�√
+2ν r
+ρ
+�ν
+Kν
+�√
+2ν r
+ρ
+�
+,
+where ρ > 0 is the range parameter, ν > 0 is the smoothness parameter, and Kν(·) is the modified
+Bessel function of second kind with order ν.
+The second estimator is the Nadaraya-Watson (N-W) kernel estimator for spatially dependent
+data discussed in Robinson (2011), which considers the following spatial regression model
+y(si) = f0(x(si)) + σ(x(si))Vi, Vi =
+∞
+�
+j=1
+aijϵj, i = 1, . . . , n,
+where f0(x) : [0, 1]d → R and σ(x) : [0, 1]d → [0, ∞) are the mean and variance functions, respec-
+tively, ϵj are independent random variables with zero mean and unit variance, and �∞
+j=1 a2
+ij = 1.
+Robinson (2011) introduces the following Nadaraya-Watson kernel estimator for f0:
+ˆf(x) = ˆν(x)
+�g(x),
+where
+�g(x) =
+1
+nhd
+n
+n
+�
+i=1
+Ki(x),
+ˆν(x) =
+1
+nhd
+h
+n
+�
+i=1
+yiKi(x),
+12
+
+with
+Ki(x) = K
+�x − x(si)
+hn
+�
+,
+and hn is a scalar, positive bandwidth sequence satisfying hn → 0 as n → ∞.
+The third estimator of f0 is based on the generalized additive model (GAM) mentioned in
+Example 1. That is, we assume that
+f0(x(s)) = Ψ
+�
+d
+�
+j=1
+gj(xj(s))
+�
+,
+where gj(·) : [0, 1] → R and Ψ(·) : R → R are some smooth functions. In this model, spatial
+dependence is not assumed.
+4.2
+Simulation Results
+To evaluate the performance of each method, we generate additional m = n/10 observations
+at new locations, treated as a test set. Similar to Chu et al. (2014), we adopt mean squared
+estimation error (MSEE) and mean squared prediction error (MSPE) to evaluate the estimation
+and prediction performance, where MSEE and MSPE are defined as
+MSEE = m−1
+m
+�
+i=1
+( �f(x(si)) − f0(x(si)))2,
+MSPE = m−1
+m
+�
+i=1
+( �f(x(si)) − y(si))2,
+and �f(x(si)) is an estimator of f0(x(si)). The mean and standard deviation of MSEE and MSPE
+over the 100 independent replicates are summarized in Tables 1 – 4.
+Tables 1 and 2 pertain to Simulation Design 1, for fixed and expanding domains, respectively.
+For each combination of the sample size n and the spatial dependence ρ, we highlight the estimator
+in boldface that yields the smallest MSEE and MSPE. Overall, GAM, GP-SVC, and N-W methods
+perform similar to each other. The proposed DNN estimator produces a smaller estimation error
+and prediction error than the others in all cases except when n = 200 and ρ = 0.1 in the fixed-
+domain case, GAM yields the smallest MSPE of 1.27. But the MSPE produced by DNN is close.
+Despite that spatial dependence has an adverse impact on the performance, when n increases (and
+D increases for the expanding-domain case), both estimation error and prediction error decrease
+as expected.
+13
+
+Table 1: Results of Simulation Design 1 with fixed domain: the averaged MSEE and MSPE over
+100 replicates (with its standard deviation in parentheses) of various methods with different n and
+ρ.
+Fixed domain
+ρ = 0.1
+ρ = 0.5
+ρ = 1
+n
+MSEE
+MSPE
+MSEE
+MSPE
+MSEE
+MSPE
+GAM
+0.92 (0.58)
+1.40 (0.69)
+0.98 (0.78)
+1.19 (0.40)
+1.05 (0.87)
+1.17 (0.38)
+n = 100
+GP-SVC
+0.87 (0.49)
+1.37 (0.67)
+0.92 (0.75)
+1.15 (0.35)
+1.01 (0.82)
+1.13 (0.36)
+N-W
+0.89 (0.52)
+1.38 (0.67)
+0.94 (0.76)
+1.16 (0.38)
+1.03 (0.85)
+1.15 (0.37)
+DNN
+0.78 (0.41)
+1.32 (0.64)
+0.82 (0.71)
+1.13 (0.35)
+0.94 (0.79)
+1.10 (0.33)
+GAM
+0.87 (0.50)
+1.26 (0.30)
+0.93 (0.72)
+1.12 (0.27)
+0.99 (0.77)
+1.08 (0.24)
+n = 200
+GP-SVC
+0.81 (0.42)
+1.34 (0.37)
+0.88 (0.74)
+1.09 (0.29)
+0.95 (0.77)
+1.09 (0.28)
+N-W
+0.84 (0.38)
+1.33 (0.41)
+0.91 (0.73)
+1.10 (0.28)
+0.93 (0.75)
+1.06 (0.25)
+DNN
+0.69 (0.32)
+1.27 (0.39)
+0.71 (0.66)
+1.06 (0.26)
+0.78 (0.68)
+1.04 (0.29)
+GAM
+0.83 (0.47)
+1.19 (0.44)
+0.88 (0.68)
+1.09 (0.20)
+0.96 (0.66)
+1.05 (0.19)
+n = 300
+GP-SVC
+0.77 (0.38)
+1.15 (0.37)
+0.86 (0.71)
+1.06 (0.21)
+0.91 (0.64)
+1.05 (0.18)
+N-W
+0.80 (0.36)
+1.13 (0.40)
+0.88 (0.70)
+1.07 (0.24)
+0.92 (0.66)
+1.06 (0.21)
+DNN
+0.58 (0.27)
+1.07 (0.34)
+0.63 (0.55)
+1.01 (0.22)
+0.69 (0.55)
+1.01 (0.25)
+We depict in Figure 1 the estimated mean functions �f(x(s)) via our method with n = 100 and
+ρ = 0.5 from the 100 replications along with the 95% pointwise confidence intervals for both fixed
+and expanding domains. Here, the 95% pointwise intervals are defined as
+�
+2−1( �f(2)(x(si)) + �f(3)(x(si))), 2−1( �f(97)(x(si)) + �f(98)(x(si)))
+�
+, i = 1, 2, . . . , n,
+where �f(k)(x(si)) is the kth smallest value of { �f[j](x(si)) : j = 1, . . . , 100}, and �f[j](x(si)) is the
+estimator of f0(x(si)) from the jth replicate.
+Tables 3 and 4 report the results for Simulation Design 2.
+For both fixed and expanding
+domains, our method performs the best among the four methods and N-W comes next. This is
+mainly because GAM and GP-SVC treat f0 to be linear and cannot handle complex interactions
+and nonlinear structures in f0.
+14
+
+Table 2: Results of Simulation Design 1 with expanding domain (i.e., D = 10, 20, 30): the averaged
+MSEE and MSPE over 100 replicates (with its standard deviation in parentheses) of various
+methods with different n and ρ.
+Expanding domain
+ρ = 0.1
+ρ = 0.5
+ρ = 1
+n
+MSEE
+MSPE
+MSEE
+MSPE
+MSEE
+MSPE
+GAM
+0.35 (0.23)
+2.06 (0.67)
+0.82 (0.38)
+1.60 (0.59)
+0.98 (0.59)
+1.42 (0.31)
+n = 100, D = 10
+GP-SVC
+0.33 (0.22)
+2.01 (0.61)
+0.78 (0.36)
+1.53 (0.55)
+0.94 (0.54)
+1.37 (0.29)
+N-W
+0.38 (0.26)
+2.03 (0.65)
+0.81 (0.40)
+1.57 (0.58)
+0.96 (0.57)
+1.39 (0.29)
+DNN
+0.26 (0.19)
+1.93 (0.55)
+0.64 (0.37)
+1.44 (0.55)
+0.76 (0.44)
+1.22 (0.26)
+GAM
+0.21 (0.14)
+1.91 (0.58)
+0.66 (0.34)
+1.51 (0.36)
+0.85 (0.44)
+1.39 (0.39)
+n = 200, D = 20
+GP-SVC
+0.18 (0.14)
+1.89 (0.54)
+0.61 (0.33)
+1.48 (0.39)
+0.81 (0.40)
+1.36 (0.41)
+N-W
+0.20 (0.17)
+1.93 (0.61)
+0.63 (0.36)
+1.47 (0.37)
+0.88 (0.48)
+1.40 (0.43)
+DNN
+0.14 (0.11)
+1.82 (0.55)
+0.43 (0.28)
+1.33 (0.32)
+0.61 (0.37)
+1.29 (0.38)
+GAM
+0.11 (0.07)
+1.72 (0.39)
+0.51 (0.29)
+1.40 (0.31)
+0.70 (0.31)
+1.30 (0.26)
+n = 300, D = 30
+GP-SVC
+0.13 (0.10)
+1.76 (0.41)
+0.56 (0.33)
+1.43 (0.36)
+0.74 (0.37)
+1.34 (0.30)
+N-W
+0.13 (0.07)
+1.77 (0.44)
+0.53 (0.30)
+1.44 (0.39)
+0.69 (0.33)
+1.29 (0.27)
+DNN
+0.07 (0.09)
+1.63 (0.38)
+0.31 (0.23)
+1.22 (0.23)
+0.43 (0.31)
+1.10 (0.19)
+5
+Data Example
+In this section, we use the proposed DNN method to analyze California Housing data that
+are publicly available from the website https://www.dcc.fc.up.pt/~ltorgo/Regression/cal_
+housing.html. After removing missing values, the dataset contains housing price information
+from n = 20433 block groups in California from the 1990 census, where a block group on average
+includes 1425.5 individuals living in a geographically compact area. To be specific, the dataset
+comprises median house values and six covariates of interest: the median age of a house, the total
+number of rooms, the total number of bedrooms, population, the total number of households, and
+the median income for households. Figure 2 displays the histograms of the six covariates, from
+which one can observe that the covariates are all right skewed except for the median age of a
+house. Thus, we first apply the logarithm transform to the five covariates and then use min-max
+normalization to rescale all the six covariates so that the data are within the range [0, 1].
+Figure 3 shows the spatial distribution of the five log transformed covariates (i.e., the total
+15
+
+Table 3: Results of Simulation Design 2 with fixed domain: the averaged MSEE and MSPE over
+100 replicates (with its standard deviation in parentheses) of various methods with different n and
+ρ.
+fixed-domain
+ρ = 0.1
+ρ = 0.5
+ρ = 1
+n
+MSEE
+MSPE
+MSEE
+MSPE
+MSEE
+MSPE
+GAM
+0.87 (0.92)
+2.81 (1.19)
+1.10 (2.40)
+2.40 (1.53)
+1.23 (1.06)
+2.16 (1.10)
+n = 100
+GP-SVC
+0.93 (0.99)
+2.93 (1.23)
+1.23 (2.54)
+2.59 (1.67)
+1.53 (1.33)
+2.37 (1.26)
+N-W
+0.73 (0.81)
+2.70 (1.15)
+1.01 (2.16))
+2.30 (1.38)
+1.09 (1.01)
+2.09 (0.98)
+DNN
+0.51 (0.60)
+2.27 (1.09)
+0.74 (1.11)
+1.90 (1.02)
+0.83 (1.19)
+1.99 (1.17)
+GAM
+0.44 (0.52)
+2.38 (0.58)
+0.75 (0.65)
+1.89 (0.42)
+0.92 (0.92)
+1.69 (0.47)
+n = 400
+GP-SVC
+0.50 (0.59)
+2.43 (0.66)
+0.82 (0.77)
+2.94 (0.47)
+1.00 (0.96)
+1.80 (0.53)
+N-W
+0.39 (0.44)
+1.99 (0.58)
+0.68 (0.70)
+1.73 (0.41)
+0.83 (0.84))
+1.56 (0.41)
+DNN
+0.22 (0.39)
+1.87 (0.49)
+0.54 (0.61)
+1.57 (0.37)
+0.68 (0.71)
+1.41 (0.37)
+GAM
+0.31 (0.40)
+2.25 (0.53)
+0.59 (0.53)
+1.81 (0.36)
+0.80 (0.79)
+1.65 (0.36)
+n = 900
+GP-SVC
+0.38 (0.44)
+2.29 (0.58)
+0.66 (0.60)
+1.88 (0.39)
+0.88 (0.83)
+1.73 (0.40)
+N-W
+0.25 (0.34)
+1.86 (0.49)
+0.51 (0.46)
+1.70 (0.31)
+0.71 (0.72))
+1.52 (0.34)
+DNN
+0.19 (0.27)
+1.70 ( 0.42)
+0.28 (0.33)
+1.49 (0.29)
+0.57 (0.59)
+1.33 (0.34)
+number of rooms, the total number of bedrooms, population, the total number of households, and
+the median income for households) and the median age of a house. We also depict in Figure 4 (the
+top panel) the map of the median house values in California. The data exhibit a clear geographical
+pattern. Home values in the coastal region, especially the San Francisco Bay Area and South Coast,
+are higher than the other regions. Areas of high home values are always associated with high
+household income, dense population, large home size, and large household, which are clustered in
+the coastal region and Central Valley. Our objective is to explore the intricate relationship between
+the median house value and the six covariates by taking into account their spatial autocorrelation.
+Same as the simulation study, we estimate the mean function f0(·) via four methods: DNN,
+GAM, GP-SVC, and N-W. To assess their performance, we compute the out-of-sample prediction
+error measured by MSPE based on 10-fold cross-validation, and the results are summarized in
+Table 5. Consistent with the observations in the simulation study, the proposed DNN method
+yields a much more accurate prediction than the others. The bottom panel of Figure 4 shows
+the estimated median house value using the DNN estimator, which exhibits a similar geographical
+16
+
+Table 4: Results of Simulation Design 2 with expanding domain (D = 10, 20, 30): the averaged
+MSEE and MSPE over 100 replicates (with its standard deviation in parentheses) of various
+methods with different n and ρ.
+fixed-domain
+ρ = 0.1
+ρ = 0.5
+ρ = 1
+n
+MSEE
+MSPE
+MSEE
+MSPE
+MSEE
+MSPE
+GAM
+0.75 (0.81)
+2.88 (1.04)
+0.81 (0.65)
+2.72 (0.94)
+0.90 (0.76)
+2.65 (0.88)
+n = 100, D = 10
+GP-SVC
+0.84 (0.88)
+2.93 (1.11)
+0.89 (0.90)
+2.80 (0.97)
+0.96 (0.83))
+2.71 (0.91)
+N-W
+0.66 (0.70)
+2.71 (1.00)
+0.71 (0.62)
+2.66 (0.90)
+0.82 (0.71))
+2.59 (0.81)
+DNN
+0.44 (0.49)
+2.15 (1.02)
+0.60 (1.03)
+1.81 (0.99)
+0.69 (1.07)
+1.76 (0.94)
+GAM
+0.32 (0.25)
+2.49 (0.44)
+0.35 (0.24)
+2.39 (0.46)
+0.40 (0.33)
+2.32 (0.51)
+n = 400, D = 20
+GP-SVC
+0.40 (0.31)
+2.55 (0.48)
+0.49 (0.29)
+2.44 (0.50)
+0.54 (0.37))
+2.40 (0.55)
+N-W
+0.28 (0.23)
+2.35 (0.41)
+0.31 (0.20)
+2.33 (0.41)
+0.35 (0.30)
+2.27 (0.47)
+DNN
+0.18 (0.20)
+2.20 (0.38)
+0.24 (0.21)
+2.29 (0.38)
+0.29 (0.24))
+2.20 (0.41)
+GAM
+0.24 (0.29)
+2.28 (0.33)
+0.27 (0.16)
+2.25 (0.27)
+0.31 (0.17)
+2.26 (0.25)
+n = 900, D = 30
+GP-SVC
+0.31 (0.32))
+2.31 (0.35)
+0.37 (0.21)
+2.17 (0.25)
+0.41 (0.20)
+2.29 (0.28)
+N-W
+0.21 (0.30)
+1.82 (0.46)
+0.26 (0.21)
+1.61 (0.28)
+0.29 (0.16)
+1.50 (0.30)
+DNN
+0.16 (0.22))
+1.71 (0.30))
+0.19 (0.20)
+1.58 (0.25)
+0.23 (0.15))
+1.45 (0.28)
+�
+��
+��
+��
+��
+��
+�
+���
+���
+���
+���
+����
+����
+���������������������
+�
+�����
+�����
+�����
+�����
+�
+����
+����
+����
+����
+����
+���������������������
+�
+����
+����
+����
+�
+���
+����
+����
+����
+����
+����
+����
+����
+������������������������
+�
+�����
+�����
+�����
+�
+����
+����
+����
+����
+����
+����
+����
+����
+����������
+�
+����
+����
+����
+����
+����
+����
+�
+���
+����
+����
+����
+����
+����
+����
+����
+��������������������������
+���
+���
+���
+���
+����
+����
+����
+�
+���
+���
+���
+���
+����
+����
+����
+����������������������������
+Figure 2: Histograms of six covariates in California housing data example.
+pattern to the observations.
+17
+
+Figure 3: The map of six covariates in California housing data example.
+Table 5: Summary of the mean squared prediction error in California housing data example.
+Methods
+GAM
+GP-SVC
+N-W
+DNN
+MSPE (×104)
+4.74
+4.23
+4.05
+3.41
+18
+
+Median age of a house
+= 50
+Total number of rooms
+ 10 
+Total number of bedrooms
+- 8
+42
+42
+42
+ 40 
+-7
+40
+40 
+ 8
+40
+- 6
+ 30
+6
+ 5
+36
+ 4
+20
+ 4
+ 3
+34
+34
+34
+-2
+32
+10
+32
+-2
+32
+-1
+125.0 122.5 120.0 117.5
+115.0
+125.0 122.5 120.0 117.5
+115.0
+125.0 122.5 120.0 117.5 115.0
+0
+Longitude
+Longitude
+Longitude
+Population
+ 10 
+Total number of households
+ 8
+Median income for households
+2.5
+42
+42
+42
+-7
+ 2.0
+40
+-8
+40
+-6
+40
+ 1.5
+ 5
+- 4
+ 1.0
+ 3
+ 0.5
+34
+34 
+34
+-2
+ 0.0
+32
+2
+32
+32
+125.0 122.5 120.0 117.5 115.0
+125.0 122.5 120.0 117.5115.0
+125.0 122.5 120.0117.5 115.0
+0.5
+Longitude
+Longitude
+0
+LongitudeFigure 4: The top panel is the map of 20433 observations and the corresponding median house
+value in California housing data example. The bottom panel is the estimated median house value.
+19
+
+ObservedMedianHouseValue
+500000
+42
+40
+400000
+38
+300000
+36
+200000
+34
+100000
+32
+0
+124
+122
+120
+118
+116
+114
+Longitude
+Estimated Median House Value
+500000
+42
+40
+400000
+38
+300000
+Latitude
+36
+200000
+34
+100000
+32
+124
+122
+120
+118
+116
+114
+LongitudeReferences
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+6
+Appendix
+6.1
+Notation and Definition
+In this paper all vectors are column vectors, unless otherwise stated. Let ∥v∥2
+2 = v⊤v for any
+vector v, and ∥f∥2 =
+��
+f(x)2dx be the L2 norm of a real-valued function f(x). For two positive
+sequences an and bn, we write an ≲ bn if there exists a positive constant c such that an ≤ cbn for all
+n, and an ≍ bn if c−1an ≤ bn ≤ can for some constant c > 1 and a sufficiently large n. Suppose that
+x = (x1, . . . , xd)⊤ is a d-dimensional vector. Let |x| = (|x1|, . . . , |xd|)⊤, |x|∞ = maxi=1,...,d |xi|,
+and |x|0 = �d
+i=1 1(xi ̸= 0). For two d−dimensional vectors x and y, we write x ≲ y if xi ≲ yi for
+i = 1, . . . , d. Let ⌊x⌋ be the largest number less than x and ⌈x⌉ be the smallest number greater
+than x. For a matrix A = (aij), let ∥A∥∞ = maxij |aij| the max norm of A, ∥A∥0 be the number
+of non-zero entries of A. Define ∥f∥∞ as the sup-norm of a real-valued function f. We use a ∧ b
+to represent the minimum of two numbers a and b, while a ∨ b is the maximum of a and b.
+Definition 1 (H¨older smoothness). A function g : Rr0 → R is said to be (β, C)-H¨older smooth for
+some positive constants β and C, if for every γ = (γ1, . . . , γr0) ∈ Nr0, the following two conditions
+hold:
+sup
+x∈Rr0
+����
+∂κg
+∂xγ1
+1 . . . ∂x
+γr0
+r0
+(x)
+���� ≤ C,
+for κ ≤ ⌊β⌋,
+and
+����
+∂κg
+∂xγ1
+1 . . . ∂x
+γr0
+r0
+(x) −
+∂κg
+∂xγ1
+1 . . . ∂x
+γr0
+r0
+(�x)
+���� ≤ C∥x − �x∥β−⌊β⌋
+2
+,
+for κ = ⌊β⌋, x, �x ∈ Rr0,
+where κ = �r0
+i=1 γi. Moreover, we say g is (∞, C)-H¨older smooth if g is (β, C)-H¨older smooth for
+all β > 0.
+6.2
+Proof of Theorem 1
+The proof of Theorem 1 requires a preliminary lemma. First, we define the δ-cover of a function
+space F as a set �F ⊂ F satisfying that following property: for any f ∈ F, there exists a �f ∈ �F
+such that ∥ �f − f∥∞ ≤ δ. Next, we define the δ-covering number of F as
+N(δ, F, ∥ · ∥∞) .= min{| �F| : �F is a δ-cover of F},
+24
+
+where |A| means the number of distinct elements in set A. In other words, N(δ, F, ∥ · ∥∞) is the
+minimal number of ∥ · ∥∞-balls with radius δ that covers F.
+Lemma 1. Suppose that f0 is the unknown true mean function in (1). Let F be a function class
+such that {f0} ∪ F ⊂ {f : [0, 1]d → [−F, F]} for some F ≥ 1. Then for all δ, ϵ ∈ (0, 1] and �f ∈ F,
+the following inequality holds:
+Rn( �f, f0) ≤(1 + ε)
+�
+inf
+˜f∈F Rn( ˜f, f0) + ∆n( �f) + 2δ
+�
+n−1 tr(Γn) + 2
+�
+n−1 tr(Γ2
+n) + 3σ
+��
++ (1 + ε)2F 2
+nε (3 log N + 1)(n−1 tr(Γ2
+n) + σ2 + 1),
+where N = N(δ, F, ∥ · ∥∞).
+Proof. Let ∆n = ∆n( �f).
+For any fixed f ∈ F, we have Ef0
+�
+n−1 �n
+i=1(y(si) − f(x(si))2 −
+inf ˜f∈F n−1 �n
+i=1(y(si) − ˜f(x(si))2�
+≥ 0. Therefore,
+Ef0
+�1
+n
+n
+�
+i=1
+(y(si) − �f(x(si))2�
+≤Ef0
+�1
+n
+n
+�
+i=1
+(y(si) − �f(x(si))2 + 1
+n
+n
+�
+i=1
+(y(si) − f(x(si))2 − inf
+˜f∈F
+1
+n
+n
+�
+i=1
+(y(si) − ˜f(x(si))2�
+=Ef0
+�1
+n
+n
+�
+i=1
+(y(si) − f(x(si)))2�
++ ∆n.
+Let ϵi = e1(si) + e2(si). Furthermore, we have
+Rn( �f, f0) = 1
+n
+n
+�
+i=1
+Ef0
+�� �f(x(si)) − f0(x(si))
+�2�
+= 1
+n
+n
+�
+i=1
+Ef0
+�� �f(x(si)) − y(si) + y(si) − f0(x(si))
+�2�
+= 1
+n
+n
+�
+i=1
+Ef0
+�� �f(x(si)) − y(si)
+�2 + ϵ2
+i + 2
+� �f(x(si)) − y(si)
+�
+ϵi
+�
+≤ 1
+n
+n
+�
+i=1
+Ef0
+�
+(y(si) − f(x(si)))2 − ϵ2
+i + 2 �f(x(si))ϵi
+�
++ ∆n
+= 1
+n
+n
+�
+i=1
+Ef0
+�
+(y(si) − f0(x(si)) + f0(x(si)) − f(x(si)))2 − ϵ2
+i + 2 �f(x(si))ϵi
+�
++ ∆n
+= 1
+n
+n
+�
+i=1
+Ef0
+�
+(f0(x(si)) − f(x(si)))2 + 2 �f(x(si))ϵi
+�
++ ∆n.
+(8)
+25
+
+Next, we will find an upper bound for Ef0
+� 2
+n
+�n
+i=1 �f(x(si))ϵi
+�
+. By the definition of the δ-
+cover of a function space F and the δ-covering number, we denote the centers of the balls by
+fj, j = 1, 2, . . . , N; and there exists fj∗ such that ∥ �f − fj∗∥∞ ≤ δ. Together with the fact that
+E
+�
+f0(x(si))ϵi
+�
+= 0, we have
+E
+�2
+n
+n
+�
+i=1
+�f(x(si))ϵi
+�
+=E
+�2
+n
+n
+�
+i=1
+� �f(x(si)) − fj∗(x(si)) + fj∗(x(si) − f0(x(si))
+�
+ϵi
+�
+≤E
+���2
+n
+n
+�
+i=1
+� �f(x(si)) − fj∗(x(si))
+�
+ϵi
+��� + E
+���2
+n
+n
+�
+i=1
+�
+fj∗(x(si)) − f0(x(si))
+�
+ϵi
+���
+≤2δ
+n E
+�
+n
+�
+i=1
+��ϵi
+��
+�
++ 2
+nE
+���
+n
+�
+i=1
+�
+fj∗(x(si)) − f0(x(si))
+�
+ϵi
+���
+.=T1 + T2.
+(9)
+It is easy to see that T1 ≤ 2δ(n−1 tr Γn + σ). For the second term T2, notice that, conditionally on
+{x(s1), . . . , x(sn)},
+ηj .=
+�n
+i=1
+�
+fj(x(si)) − f0(x(si))
+�
+ϵi
+�
+a⊤
+j Γnaj + nσ2∥fj − f0∥2
+n
+follows N(0, 1) where aj = (fj(x(s1))−f0(x(s1)), . . . , fj(x(sn))−f0(x(sn)))⊤, ∥f∥2
+n = n−1 �n
+i=1 f(x(si))2.
+From Lemma C.1 of Schmidt-Hieber (2020), Ef0
+�
+maxj=1,...,N η2
+j|x(s1), . . . , x(sn)
+�
+≤ 3 log N + 1.
+Consequently,
+T2 =2
+nE
+���ηj∗
+�
+a⊤
+j∗Γnaj∗ + nσ2∥fj∗ − f0∥2
+n
+���
+≤2
+nE
+�
+|ηj∗|
+�
+(tr(Γ2
+n) + nσ2)∥fj∗ − f0∥2
+n
+�
+≤2
+�
+tr(Γ2
+n) + nσ2
+n
+E
+�
+|ηj∗|(∥ ˆf − f0∥n + δ)
+�
+≤2
+�
+tr(Γ2
+n) + nσ2
+n
+�
+3 log N + 1
+��
+Rn( �f, f0) + δ
+�
+Together with (8) and (9), we have
+Rn( �f, f0) ≤Rn(f, f0) + ∆n + 2δ(n−1 tr Γn + σ) + 2
+�
+tr(Γ2
+n) + nσ2
+n
+�
+3 log N + 1
+��
+Rn( �f, f0) + δ
+�
+.
+26
+
+If log N ≤ n, then
+Rn( �f, f0) ≤Rn(f, f0) + ∆n + 2δ
+�
+n−1 tr(Γn) + σ + 2
+�
+n−1 tr(Γ2
+n) + σ2
+�
++ 2
+�
+tr(Γ2
+n) + nσ2
+n
+�
+3 log N + 1
+�
+Rn( �f, f0).
+Applying the inequality (43) in Schmidt-Hieber (2020), we have, for any 0 < ε ≤ 1,
+Rn( �f, f0) ≤(1 + ε)
+�
+Rn(f, f0) + ∆n + 2δ
+�
+n−1 tr(Γn) + σ + 2
+�
+n−1 tr(Γ2
+n) + σ2
+��
++ (1 + ε)2
+ε
+1
+n2(3 log N + 1)(tr(Γ2
+n) + nσ2)
+≤(1 + ε)
+�
+Rn(f, f0) + ∆n + 2δ
+�
+n−1 tr(Γn) + 2
+�
+n−1 tr(Γ2
+n) + 3σ
+��
++ (1 + ε)2F 2
+nε (3 log N + 1)(n−1 tr(Γ2
+n) + σ2 + 1).
+For log N > n, Rn( �f, f0) = 1
+n
+�n
+i=1 Ef0
+�� �f(x(si)) − f0(x(si))
+�2�
+≤ 4F 2 and
+(1 + ε)
+�
+Rn(f, f0) + ∆n + 2δ
+�
+n−1 tr(Γn) + 2
+�
+n−1 tr(Γ2
+n) + 3σ
+��
++ (1 + ε)2F 2
+nε (3 log N + 1)(n−1 tr(Γ2
+n) + σ2 + 1)
+>2F 2
+n (3n + 1) > 6F 2.
+Thus,
+Rn( �f, f0) ≤(1 + ε)
+�
+Rn(f, f0) + ∆n + 2δ
+�
+n−1 tr(Γn) + 2
+�
+n−1 tr(Γ2
+n) + 3σ
+��
++ (1 + ε)2F 2
+nε (3 log N + 1)(n−1 tr(Γ2
+n) + σ2 + 1).
+Since the above inequality holds true for any f ∈ F, we can prove the result by letting f =
+arginf ˜f∈F Rn( ˜f, f0).
+Proof of Theorem 1: It follows from Lemma 5 and Remark 1 of Schmidt-Hieber (2020) that
+log N = log N(δ, F(L, p, τ, F), ∥ · ∥∞) ≤(1 + τ) log(25+2Lδ−1(L + 1)τ 2Ld2).
+Because F ≥ 1 and 0 < ε ≤ 1, we have
+Rn( �f, f0) ≲(1 + ε)
+�
+inf
+˜f∈F(L,p,τ,F) ∥ ˜f − f0∥2
+∞ + ∆n( �f) + ςn,ε,δ
+�
+,
+27
+
+where
+ςn,ε,δ ≍1
+ε
+�
+δ
+�
+n−1 tr(Γn) + 2
+�
+n−1 tr(Γ2
+n) + 3σ
+�
++ τ
+n (log(L/δ) + L log τ) (n−1 tr(Γ2
+n) + σ2 + 1)
+�
+.
+□
+6.3
+Proof of Theorem 2
+Lemma 2. For any f : Rd → R ∈ CS(L∗, r, ˜r, β, a, b, C), m ∈ Z+, and N ≥ maxi=0,...,L∗(βi +
+1)˜ri ∨ ( ˜Ci + 1)e˜ri, there exists a neural network
+f∗ ∈ F(L, (d, 6ηN, . . . , 6ηN, 1),
+L∗
+�
+i=0
+ri+1(τi + 4), ∞),
+such that
+∥f∗ − f∥∞ ≤ CL∗
+L∗−1
+�
+l=0
+(2Cl)βl+1
+L∗
+�
+i=0
+�
+(2 ˜Ci + 1)(1 + ˜r2
+i + β2
+i )6˜rN2−m + ˜Ci3βiN − βi
+˜ri
+��L∗
+l=i+1 βl∧1
+,
+where
+˜Ci =
+i
+�
+k=0
+Ck
+bk − ak
+bk+1 − ak+1
+, i = 0, . . . , L∗ − 1,
+˜CL∗ =
+L∗
+�
+k=0
+Ck
+bk − ak
+bk+1 − ak+1
++ bL∗ − aL∗
+L = 3L∗ +
+L∗
+�
+i=0
+Li,
+with Li = 8 + (m + 5)(1 + ⌈log2(˜ri ∨ βi)⌉),
+τi ≤ 141(˜ri + βi + 1)3+˜riN(m + 6), i = 0, . . . , L∗,
+η =
+max
+i=0,...,L∗(ri+1(˜ri + ⌈βi⌉)).
+Proof. By definition, we write f(z) as
+f(z) = gL∗ ◦ . . . ◦ g1 ◦ g0(z),
+for z ∈ [a0, b0]r0,
+where gi = (gi,1, . . . , gi,ri+1)⊤ : [ai, bi]ri → [ai+1, bi+1]ri+1 for some |ai|, |bi| ≤ Ci and the functions
+gi,j : [ai, bi]˜ri → [ai+1, bi+1] are (βi, Ci)-H¨older smooth and rL∗+1 = 1. For i = 0, . . . , L∗ − 1, the
+domain and range of gi are [ai, bi]ri and [ai+1, bi+1]ri+1, respectively. First of all, we will rewrite f
+as the composition of the functions hi := (hi,1, . . . , hi,ri+1)⊤ whose domain and range are [0, 1]ri
+and [0, 1]ri+1 which are constructed via linear transformation. That is, we define
+hi(z) := gi((bi − ai)z − ai+1)
+bi+1 − ai+1
+,
+for z ∈ [0, 1]ri, i = 0, . . . , L∗ − 1,
+hL∗(z) := gL∗((bL∗ − aL∗)z + aL∗),
+for z ∈ [0, 1]rL∗.
+28
+
+Therefore the following equality holds
+f(z) = gL∗ ◦ . . . ◦ g1 ◦ g0(z) = hL∗ ◦ . . . ◦ h1 ◦ h0( z − a0
+b0 − a0
+),
+for z ∈ [a0, b0]r0.
+Since gi,j : [ai, bi]˜ri → [ai+1, bi+1] are all (βi, Ci)-H¨older smooth, it follows that hi,j : [0, 1]˜ri → [0, 1]
+are all (βi, ˜Ci)-H¨older smooth as well, where ˜Ci is a constant only depending on a, b, and C, i.e.,
+˜Ci = �i
+k=0 Ck
+bk−ak
+bk+1−ak+1 for i = 0, . . . , L∗ − 1, and ˜CL∗ = �L∗
+k=0 Ck
+bk−ak
+bk+1−ak+1 + bL∗ − aL∗.
+By Theorem 5 of Schmidt-Hieber (2020), for any integer m ≥ 1 and N ≥ maxi=0,...,L∗(βi +
+1)˜ri ∨ ( ˜Ci + 1)e˜ri, there exists a network
+˜hi,j ∈ F(Li, (˜ri, 6(˜ri + ⌈βi⌉)N, . . . , 6(˜ri + ⌈βi⌉)N, 1), τi, ∞),
+with Li = 8 + (m + 5)(1 + ⌈log2(˜ri ∨ βi)⌉) and τi ≤ 141(˜ri + βi + 1)3+˜riN(m + 6), such that
+∥˜hi,j − hi,j∥∞ ≤ (2 ˜Ci + 1)(1 + ˜r2
+i + β2
+i )6˜riN2−m + ˜Ci3βiN − βi
+˜ri .
+Note that the value of ˜hi,j is (−∞, ∞). So we define h∗
+i,j := σ(−σ(−˜hi,j + 1) + 1) by adding
+two more layers σ(1 − x) to restrict h∗
+i,j into the interval [0, 1], where σ(x) = max(0, x). This
+introduces two more layers and four more parameters. By the fact that hi,j ∈ [0, 1], we have
+h∗
+i,j ∈ F(Li + 2, (˜ri, 6(˜ri + ⌈βi⌉)N, . . . , 6(˜ri + ⌈βi⌉)N, 1), τi + 4, ∞) and
+∥h∗
+i,j − hi,j∥∞ ≤ ∥˜hi,j − hi,j∥∞ ≤ (2 ˜Ci + 1)(1 + ˜r2
+i + β2
+i )6˜riN2−m + ˜Ci3βiN − βi
+˜ri .
+We further parallelize all (h∗
+i,j)j=1,...,ri+1 together, obtaining h∗
+i := (h∗
+i,1, . . . , h∗
+i,ri+1)⊤ ∈ F(Li +
+2, (ri, 6ri+1(˜ri + ⌈βi⌉)N, . . . , 6ri+1(˜ri + ⌈βi⌉)N, ri+1), ri+1(τi + 4), ∞). Moreover, we construct the
+composite network f∗ := h∗
+L∗ ◦. . .◦h∗
+1◦h∗
+0 ∈ F(3L∗+�L∗
+i=0 Li, (r0, 6ηN, . . . , 6ηN, 1), �L∗
+i=0 ri+1(τi+
+4), ∞), where η = maxi=0,...,L∗(ri+1(˜ri + ⌈βi⌉)).
+29
+
+By Lemma 3 in Schmidt-Hieber (2020),
+∥f − f∗∥∞ =∥hL∗ ◦ . . . ◦ h1 ◦ h0 − h∗
+L∗ ◦ . . . ◦ h∗
+1 ◦ h∗
+0∥∞
+≤CL∗
+L∗−1
+�
+l=0
+(2Cl)βl+1
+L∗
+�
+i=0
+∥|hi − h∗
+i |∞∥
+�L∗
+l=i+1 βl∧1
+∞
+≤CL∗
+L∗−1
+�
+l=0
+(2Cl)βl+1
+L∗
+�
+i=0
+�
+(2 ˜Ci + 1)(1 + ˜r2
+i + β2
+i )6˜rN2−m + ˜Ci3βiN − βi
+˜ri
+��L∗
+l=i+1 βl∧1
+≤CL∗
+L∗−1
+�
+l=0
+(2Cl)βl+1
+L∗
+�
+i=0
+((2 ˜Ci + 1)(1 + ˜r2
+i + β2
+i )6˜rN2−m)
+�L∗
+l=i+1 βl∧1
++ CL∗
+L∗−1
+�
+l=0
+(2Cl)βl+1
+L∗
+�
+i=0
+( ˜Ci3βiN − βi
+˜ri )
+�L∗
+l=i+1 βl∧1.
+Proof of Theorem 2: By Theorem 1 with δ = n−2 and ε = 1, it follows that
+Rn( �flocal, f0) ≲
+inf
+˜f∈F(L,p,τ,F) ∥ ˜f − f0∥2
+∞ + (tr(Γ2
+n) + n)τ(log(Ln2) + L log τ)
+n2
++ ∆n( �flocal).
+Next, we need to analyze the first term. Since f0 ∈ CS(L∗, r, ˜r, β, a, b, C), by Lemma 2, for any
+m > 0, there exists a neural network
+f∗ ∈ F(L, (d, N, . . . , N, 1), τ, ∞),
+with L ≍ m, N ≥ 6η maxi=0,...,L∗(βi + 1)˜ri ∨ ( ˜Ci + 1)e˜ri, η = maxi=0,...,L∗(ri+1(˜ri + ⌈βi⌉)), τ ≲ mN,
+such that
+∥f∗ − f0∥∞ ≲
+L∗
+�
+i=0
+(N2−m)
+�L∗
+l=i+1 βl∧1 + (N − βi
+˜ri )
+�L∗
+l=i+1 βl∧1
+≲
+L∗
+�
+i=0
+(N2−m)
+�L∗
+l=i+1 βl∧1 + N −
+β∗
+i
+˜ri
+≲ (N2−m)
+�L∗
+l=1 βl∧1 + N − β∗
+r∗ ,
+(10)
+where recall that β∗ = β∗
+i∗ and r∗ = ˜ri∗.
+For simplicity, we let p = (d, N, . . . , N, 1).
+This
+means there exists a sequence of networks (fn)n such that for all sufficiently large n, ∥fn −
+f0∥∞ ≲ (N2−m)
+�L∗
+l=1 βl∧1 + N − β∗
+r∗ and fn ∈ F(L, p, τ, ∞). Next define `f := fn(∥f0∥∞/∥fn∥∞ ∧ 1) ∈
+30
+
+F(L, p, τ, F), F ≥ maxi=0,...,L∗(Ci, 1), and it is obvious that ∥ `f − f0∥∞ ≲ (N2−m)
+�L∗
+l=1 βl∧1 + N − β∗
+r∗ .
+Then it follows that
+inf
+˜f∈F(L,p,τ,F) ∥ ˜f − f0∥∞ ≲ ∥ `f − f0∥∞ ≲ (N2−m)
+�L∗
+l=1 βl∧1 + N − β∗
+r∗ .
+(11)
+By combining (10) and the fact that τ ≲ LN, the proof is completed. □
+31
+

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+Electron heating and radiation in high aspect ratio sub-micron plasma generated by
+an ultrafast Bessel pulse within a solid dielectric
+Kazem Ardaneh∗
+FEMTO-ST Institute, Univ.
+Franche-Comt´e, CNRS,
+15B avenue des Montboucons, 25030 Besan¸con cedex, France. and
+Sorbonne University, Pierre and Marie Curie Campus, 4 place Jussieu, 75252 Paris Cedex 5, France
+Remo Giust, Pierre-Jean Charpin, Benoit Morel and Francois Courvoisier†
+FEMTO-ST Institute, Univ.
+Franche-Comt´e, CNRS,
+15B avenue des Montboucons, 25030 Besan¸con cedex, France.
+This preprint has not undergone peer review.
+The Version of Record of this article
+is published in The European Physical Journal Special Topics, and is available online at
+https://doi.org/10.1140/epjs/s11734-022-00751-y
+Full reference: K. Ardaneh, R. Giust, P.-J. Charpin, B. Morel and F. Courvoisier ” Electron
+heating and radiation in high aspect ratio sub-micron plasma generated by an ultrafast Bessel
+pulse within a solid dielectric ”, The European Physical Journal Special Topics, (2022).
+DOI:
+10.1140/epjs/s11734-022-00751-y
+When propagating inside dielectrics, an ultrafast Bessel beam creates a high aspect-ratio cylinder
+of plasma with nanometric diameter that extends over several tens of micrometers to centimeters.
+We analyze the interaction between the intense ultrafast laser pulse and the plasma rod using
+particle-in-cell simulations. We show that electrons are heated and accelerated up to keV energies
+via transit acceleration inside the resonance lobes in the vicinity of the critical surface and compute
+their radiation pattern.
+INTRODUCTION
+Ultrafast lasers are ideal tools to deposit energy within
+the bulk of transparent materials [1]. This has applica-
+tions for laser micromachining, for the generation of new
+material phases as well as for the generation of warm
+dense matter. Thanks to the nonlinear ionization, the
+infrared radiation of the laser can generate, early in the
+pulse, a plasma of electrons and holes in the bulk of trans-
+parent dielectrics [2]. The interaction of the trailing part
+of the laser pulse can heat the plasma if proper conditions
+are met. Then, depending on the energy density that has
+been deposited within the plasma, phase change can oc-
+cur, even at sub-picosecond time scale via non-thermal
+melting if the ionization rate of the solid is sufficiently
+high (only few %) [3, 4]. In this case, the material can
+quickly reach the warm dense matter regime, in which
+the material is at solid density but with temperatures on
+the order of 1-10 eV [5, 6] . This is a challenging state
+still under study which is relevant for the modeling of
+many astrophysical objects [7]. The phase change trig-
+gers also a series of physical phenomena in the material
+such as shockwave emission, void formation [8], densifica-
+tion around the void [9]. In the densified regions around
+voids formed within the bulk, the extreme temperatures
+and pressures reached within a short time can lead to the
+formation of new material phases as it has been observed
+in sapphire and silicon [10, 11]. Finally, the formation of
+voids inside the material has useful applications for laser
+cutting or drilling of transparent materials. High-speed
+(typ. 1 m/s) cutting of glass with the stealth dicing tech-
+nology is one of the most relevant examples [12–15].
+In these three domains of applications, it is clear that
+a challenge is to reach the largest energy density as
+possible over the largest volume possible.
+However, it
+is well-known that nonlinear filamentation of Gaussian
+beams prevents reaching extreme energy densities inside
+dielectrics. In contrast, we recently demonstrated that it
+is possible with Bessel beams [16].
+Zeroth-order Bessel beams, also called ”diffraction-
+free” beams, constitute a propagation-invariant solution
+to the wave equation [17]. They are featured by a conical
+flow of light directed toward the optical axis. The conical
+interaction creates an interference pattern, characterized
+by an intense central lobe surrounded by several other
+circular lobes of lower intensity. Importantly, when prop-
+agating inside transparent solids, an ultrafast laser pulse
+shaped as a Bessel beam can generate a nano-plasma rod
+with a length that is adjustable independently of the di-
+ameter. Recent work has shown that the length of this
+arXiv:2301.02408v1  [physics.plasm-ph]  6 Jan 2023
+
+2
+plasma rod can be scaled from tens of micrometers to
+1 cm [15].
+We have recently demonstrated that 100 fs Bessel
+beams can generate elongated plasma rods with over crit-
+ical plasma density in sapphire Al2O3 [16], in the regime
+corresponding to the formation of high aspect ratio nano-
+voids [18].
+In contrast with the case of the Gaussian
+beam, all the pulse energy in the Bessel beam impinges
+with a relatively large incidence angle toward the plasma
+rod generated early in the pulse along the optical axis.
+This configuration is ideal to trigger resonance absorp-
+tion. In reference [16], we have demonstrated the occur-
+rence of resonance absorption inside sapphire by com-
+paring experimental results to particle-in-cell (PIC) sim-
+ulations. The strong energy transfer between the laser
+wave to the core of the plasma opens the possibility to
+reach the warm dense matter regime and to produce
+temperatures on the order of 10 eV. This explains the
+opening of high aspect ratio nano channels inside several
+dielectrics upon Bessel beam femtosecond illumination.
+Importantly, the conical geometry of Bessel beams is in-
+variant along the propagation. This implies that results
+obtained with Bessel beams of only several tens of mi-
+crometers in length can be extrapolated to several cen-
+timeters.
+A key question is the understanding of the micro-
+physics of the interaction of the Bessel beam with the
+sub-micron plasma, at moderate intensities (typically
+1014 W/cm2).
+For this, we have used Particle-In-Cell
+(PIC) simulations to investigate the electron plasma wave
+generation, particle heating and acceleration, as well as
+the radiation emitted by the accelerated particles.
+SIMULATIONS
+Our PIC simulations are based on the interaction of a
+Bessel-Gauss beam [19] with a preformed plasma. The
+100 fs laser pulse, with a central wavelength of 800 nm,
+is polarized along x-direction. We assume that nonlinear
+ionization has produced early in the pulse a plasma and
+we model the interaction of the most intense part of the
+laser pulse with this pre-plasma. We used the numerical
+code EPOCH [20], without ionization, and the numerical
+scheme is detailed in reference [21]. We maintained the
+numerical heating at a negligible level over the duration
+of the simulation (320 fs). The pre-plasma is a plasma
+rod extending over the whole longitudinal length of the
+box (because of the invariance of the Bessel beam), and
+its transverse cross-section is elliptically-shaped.
+The
+profile is Gaussian and is the same as the one match-
+ing our experimental results, as shown in reference [16]:
+the critical radius is 190 nm along the polarization di-
+rection (x axis) and 380 nm in the other direction, as
+shown in Fig. 1(a). The peak intensity of the pulse is
+6×1014W/cm2, for a pulse duration of 100 fs.
+FIELD AMPLIFICATION AND PARTICLE
+ACCELERATION
+Figure 1(b) shows the evolution of the Ex component
+of the electric field in time, on a segment placed along the
+x direction at the propagation distance corresponding to
+the highest intensity reached in the Bessel-Gauss beam
+[22]. We first observe the field enhancement in the region
+where the permittivity decreases because of the presence
+of plasma ( |x| < 190 nm). A strong field amplification
+occurs on the critical surfaces, that are indicated with
+dashed lines in Fig. 1(a). The field reaches a maximum
+value of 1 GV/cm. This corresponds to an amplification
+factor of approximately 7 in comparison with the input
+laser field.
+At the critical surface, the resonance absorption takes
+place.
+The wave conversion phenomenon creates elec-
+tron plasma waves. We plot in Fig. 2(a) the density of
+electrons at the peak of the pulse, in x − z plane. The
+white line shows the contour at the critical density. We
+see plasma oscillations taking place. To allow a clearer
+visualization of the plasma waves, we computed the elec-
+trostatic field, that is shown in Fig. 2(b). It has been de-
+rived from the density ρ using Gauss’s law in the Fourier
+k-space, EES
+x
+= −jkρ(k)/k2/ϵ0.
+We can observe that, in the sub- critical region, out-
+side the plasma, the electron plasma waves are quickly
+damped. This arises from the efficient Landau damping.
+In the over-critical region, the field oscillations are due to
+electron sound waves [21, 23]. They penetrate relatively
+deeply into the overcritical region and we observe that
+they are curved. The propagation of these waves into the
+overcritical region is attributed to the fact that the tem-
+perature is highly inhomogeneous in the plasma. This
+is also the reason of the variation of the spatial period
+along x, hence the variation of the apparent curvature of
+the fringes. This structure will have a strong impact on
+the acceleration of electrons as we will see below.
+The heating of the particles has been investigated in
+Ref. [16]. In summary, the wave particle energy exchange
+is very efficient: we observe the heating of the electron
+population mainly around the critical surfaces. In Fig.
+2(c), we show the particle distribution in the phase space
+x − px after the pulse (135 fs). We evaluated the main
+component temperature to be 1.3 eV, and a hot electrons
+component at 70 eV. In addition, we observe in Fig. 2(c),
+two tails expanding outwards, at high momentum, out-
+side the plasma.
+These tails correspond to highly accelerated particles
+with energies up to 7 keV. Figure 1 provides a glimpse on
+the physical mechanism of the acceleration. We have se-
+lected out 1000 of the most energetic particles and have
+traced their trajectories.
+In Fig.
+1(b), we have plot-
+ted a selection of 8 representative trajectories. Two of
+them show highly accelerated particles that escape from
+the plasma.
+The other 6 are accelerated by the same
+
+3
+FIG. 1. (a) Cross section of the 2D density profile of the plasma. The black solid line shows the density profile at y = 0
+along x-direction. The critical density nc is 1.7 × 1021 cm−3 (b) Ex component of the electric field as a function of time and x
+position. The solid colored lines correspond to the projection in the plane of 8 particle trajectories. Their energy is indicated
+using the color code. (c-h) Zoom-in views of the acceleration of the electrons as they leave the plasma, together with the exact
+value of the field Ex(x, y = yp, z = zp, t) sampled at the particle position (xp,yp,zp, t).
+FIG. 2.
+(a) Plasma density at the time corresponding to the peak intensity in the plane y = 0, which is parallel to the
+polarization direction. The white solid line shows the contour at the critical density(b) Longitudinal component of the electric
+field, which allows the visualization of the plasma density waves. (c) x − px phase space of the electron population at a time
+135 fs after the peak of the pulse.
+
+4
+(c)
+(d)
+(e)
+[keV]
+2.7 fs
+2.7 fs
+2.7 fs
+wu
+1 -
+0.
+n[nc]
+0
+2
+4
+1
+0.5
+(a)
+(b)
+0.2
+Ex[GV/cm]
+x[μm]
+0
+0.0
+-0.2
+-0.5
+-1
+0
+1
+-100
+-50
+0
+50
+-1
+100
+150
+y[μm]
+t - tc [fs]
+5
+(f)
+(a)
+(h)
+4
+2 
+n
+1 
+2.7 fs
+2.7 fs
+2.7 fs
+00.5
+5
+100
+20
+(e)
+(c)
+[μm]
+n[nc]
+3
+0.0
+50
+15
+2
++
+[xd
+[keV/c]
+1
+1'XIN 601
+-0.5
+0
+0
+10
+10
+15
+0.5
+0.5
+(b)
+α
+[μm]
+-50
+5
+0.0
+0.0
+-0.5
+-0.5
+-100
+0
+15
+10
+15
+-5
+z[μm]
+x[μm]4
+FIG. 3.
+(left)The total energy radiated per unit solid an-
+gle per unit frequency from the accelerated electrons. (right)
+Configuration of the angles. The laser polarization is along
+the x axis
+mechanism, but remain trapped by a static electric field
+generated by a double layer formation that will be ex-
+plained later. The acceleration mechanism is the same
+in all cases: in the different sub-figures 1(c-h), we see
+that the particles undergo transit acceleration, which oc-
+curs when a particle travels through a highly non-uniform
+electromagnetic field [24–26]. The electrons gain energy
+by riding on the electron plasma wave, that is curved
+in the x − t space. The effective acceleration occurs on
+a distance of less than 60 nm, when crossing the high
+resonance field. Depending on the exact position of the
+particle with respect to the plasma wave, the acceleration
+is more or less efficient. We see that the curvature of the
+plasma wave is a key to obtain a progressive acceleration
+of the particle while it remains on the peak of the wave.
+Particle acceleration and heating transfer a fraction of
+the electrons away from the main plasma, as it is also
+apparent on the x − px representation of Fig. 2(c). This
+generates a so-called double layer on either sides of the
+plasma. These double layers are apparent in Fig. 1 be-
+cause they generate a static E-field that is superimposed
+to the laser pulse. Importantly, this static field has an
+amplitude that is as high as the resonance field of the
+laser pulse itself (on the order of GV/cm). This static E-
+field even remains after the laser pulse has vanished. Its
+damping is governed by the collisions inside the plasma.
+RADIATION PATTERN
+One of the major advantages of PIC codes is the pos-
+sibility to access the full information about the parti-
+cle dynamics, e.g., the position and the momentum as
+a function of time. If this information can be retrieved
+and stored for a number of particles, it is then feasible
+to post-process the radiation associated with a partic-
+ular set of particles. The radiation diagnostic uses the
+information from the particle trajectories, position and
+momentum over time, and determines the energy being
+radiated by an accelerated charged particle.
+Let us consider a particle at position r0 (t) at time
+t.
+At the same time, we observe the radiated electro-
+magnetic fields from the particle at position r. Due to
+the finite velocity of light, we observe the particle at an
+earlier position r0 (t′) where it was at the retarded time
+t′ = t − R (t′) /c, where R (t′) = |r − r0 (t′) | is the dis-
+tance from the charged particle (at the retarded time t′ )
+to the observer. Using the Li´enard-Wiechert potentials,
+the total energy W radiated per unit solid angle per unit
+frequency from a charged particle moving with instanta-
+neous velocity β = v/c under acceleration ˙β = a/c can
+be expressed as [27]:
+d2W
+dωdΩ ∝
+�����
+� ∞
+−∞
+ˆn × [(ˆn − β) × ˙β]
+(1 − β · ˆn)2
+eiω(t−ˆn·r(t)/c)dt
+�����
+2
+(1)
+Here, n = R (t′) /|R (t′) | is a unit vector that points
+from the particle retarded position towards the observer.
+The observer’s viewing angle is set by the choice of
+n (ˆx sin θ cos φ + ˆy sin θ sin φ + ˆz cos θ).
+Figure 3 shows the total radiated energy from an en-
+semble of 100 randomly selected electrons traced in the
+simulations that we computed using Eq. (1). The distri-
+bution is plotted as a function of the angles (θ, φ) in Fig.
+3(left). The forward direction corresponds to values of
+θ below 90◦. We observe that the forward signal shows
+maxima at (θ, φ) ≈ (0, π/2) and (0, 3π/2), perpendicu-
+lar to the electron acceleration in the x−direction, which
+indeed corresponds to the pump laser polarization direc-
+tion. The emission follows the well-known power distri-
+bution per solid angle Ω emitted by a single particle [27]:
+dP
+dΩ ∝
+| ˙β|2
+(1 − β cos θ)3
+�
+1 −
+sin2 θ cos2 φ
+γ2(1 − β cos θ)2
+�
+(2)
+where γ is the Lorentz factor.
+Noticeably, the power shows a much higher signal in
+the angles corresponding to the forward direction than for
+the backward direction (90◦ < θ ≤ 180◦). This behaviour
+could be explained by the coherence of the phases of the
+dipole moments induced along the plasma rod. A more
+detailed analysis of the radiation spectrum, out of the
+scope of the present article, shows the presence of second
+harmonic generation and of THz radiation in the forward
+direction.
+In conclusion, we have investigated the interaction be-
+tween a moderately intense laser pulse shaped as a Bessel
+beam with a nanoplasma rod using particle-in-cell sim-
+ulations. We have demonstrated that resonance absorp-
+tion generate plasma waves inside plasma. These plasma
+
+dW/dw/dQ[a. u]
+3.7e-07
+1.5e-06
+2.7e-06
+3.8e-06
+5.0e-06
+180
+Z1
+150
+120
+Y
+90
+60
+30-
+0
+0
+60
+120
+180
+240
+300
+360
+[。]Φ5
+waves are highly damped in the sub- critical region while
+plasma sound waves can propagate over several hundreds
+of nanometers inside the overcritical plasma. The anal-
+ysis of the trajectories of the most energetic particles
+shows that the main acceleration mechanism is transit
+acceleration. It occurs at the critical layer when parti-
+cles are trapped inside a plasma wave and gain energy
+until they are released at the critical surface. Because of
+the heating of the electron gas, two double layers form on
+either sides of the nano-plasma. The most energetic par-
+ticles, with energies up to 7 keV can escape the plasma
+while part of the hot electrons remain trapped by the
+potential well due to the electrostatic field of the dou-
+ble layer. Overall, our results enable us to gain insights
+into the micro physics of the laser-plasma interaction
+that this relevant for the understanding of the different
+mechanisms of the deposition of the femtosecond laser
+pulse energy inside dielectrics. Our results reveal a rich
+physics which can be exploited in several fields of appli-
+cations: laser-matter interaction, laser micro-machining,
+warm dense matter and high energy density physics in-
+side solids, as well as the generation of electrostatic fields
+and terahertz radiation.
+Acknowledgments :
+The research leading to these
+results has received funding from the European Re-
+search Council (ERC) under the European Union’s Hori-
+zon 2020 research and innovation program (grant agree-
+ment No 682032-PULSAR), R´egion Bourgogne Franche-
+Comt´e, I-SITE BFC project (contract ANR-15-IDEX-
+0003), and the EIPHI Graduate School (ANR-17-EURE-
+0002). We acknowledge the support of PRACE HPC re-
+sources under the Project ”PULSARPIC” (PRA19 4980
+and RA5614), and GENCI resources under projects
+A0070511001 and A0090511001.
+Data availability statement: Data will be made avail-
+able on reasonable request.
+∗ [email protected]
+† [email protected]
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+

9NE0T4oBgHgl3EQffwBQ/content/tmp_files/load_file.txt

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+page_content='Electron heating and radiation in high aspect ratio sub-micron plasma generated by  an ultrafast Bessel pulse within a solid dielectric  Kazem Ardaneh∗  FEMTO-ST Institute, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  Franche-Comt´e, CNRS,  15B avenue des Montboucons, 25030 Besan¸con cedex, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' and  Sorbonne University, Pierre and Marie Curie Campus, 4 place Jussieu, 75252 Paris Cedex 5, France  Remo Giust, Pierre-Jean Charpin, Benoit Morel and Francois Courvoisier†  FEMTO-ST Institute, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  Franche-Comt´e, CNRS,  15B avenue des Montboucons, 25030 Besan¸con cedex, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  This preprint has not undergone peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  The Version of Record of this article  is published in The European Physical Journal Special Topics, and is available online at  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='1140/epjs/s11734-022-00751-y  Full reference: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Ardaneh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Giust, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Charpin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Morel and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Courvoisier ” Electron  heating and radiation in high aspect ratio sub-micron plasma generated by an ultrafast Bessel  pulse within a solid dielectric ”, The European Physical Journal Special Topics, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  DOI:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='1140/epjs/s11734-022-00751-y  When propagating inside dielectrics, an ultrafast Bessel beam creates a high aspect-ratio cylinder  of plasma with nanometric diameter that extends over several tens of micrometers to centimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  We analyze the interaction between the intense ultrafast laser pulse and the plasma rod using  particle-in-cell simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We show that electrons are heated and accelerated up to keV energies  via transit acceleration inside the resonance lobes in the vicinity of the critical surface and compute  their radiation pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  INTRODUCTION  Ultrafast lasers are ideal tools to deposit energy within  the bulk of transparent materials [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' This has applica-  tions for laser micromachining, for the generation of new  material phases as well as for the generation of warm  dense matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Thanks to the nonlinear ionization, the  infrared radiation of the laser can generate, early in the  pulse, a plasma of electrons and holes in the bulk of trans-  parent dielectrics [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The interaction of the trailing part  of the laser pulse can heat the plasma if proper conditions  are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Then, depending on the energy density that has  been deposited within the plasma, phase change can oc-  cur, even at sub-picosecond time scale via non-thermal  melting if the ionization rate of the solid is sufficiently  high (only few %) [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' In this case, the material can  quickly reach the warm dense matter regime, in which  the material is at solid density but with temperatures on  the order of 1-10 eV [5, 6] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' This is a challenging state  still under study which is relevant for the modeling of  many astrophysical objects [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The phase change trig-  gers also a series of physical phenomena in the material  such as shockwave emission, void formation [8], densifica-  tion around the void [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' In the densified regions around  voids formed within the bulk, the extreme temperatures  and pressures reached within a short time can lead to the  formation of new material phases as it has been observed  in sapphire and silicon [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Finally, the formation of  voids inside the material has useful applications for laser  cutting or drilling of transparent materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' High-speed  (typ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' 1 m/s) cutting of glass with the stealth dicing tech-  nology is one of the most relevant examples [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  In these three domains of applications, it is clear that  a challenge is to reach the largest energy density as  possible over the largest volume possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  However, it  is well-known that nonlinear filamentation of Gaussian  beams prevents reaching extreme energy densities inside  dielectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' In contrast, we recently demonstrated that it  is possible with Bessel beams [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  Zeroth-order Bessel beams, also called ”diffraction-  free” beams, constitute a propagation-invariant solution  to the wave equation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' They are featured by a conical  flow of light directed toward the optical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The conical  interaction creates an interference pattern, characterized  by an intense central lobe surrounded by several other  circular lobes of lower intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Importantly, when prop-  agating inside transparent solids, an ultrafast laser pulse  shaped as a Bessel beam can generate a nano-plasma rod  with a length that is adjustable independently of the di-  ameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Recent work has shown that the length of this  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='02408v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='plasm-ph]  6 Jan 2023  2  plasma rod can be scaled from tens of micrometers to  1 cm [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  We have recently demonstrated that 100 fs Bessel  beams can generate elongated plasma rods with over crit-  ical plasma density in sapphire Al2O3 [16], in the regime  corresponding to the formation of high aspect ratio nano-  voids [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  In contrast with the case of the Gaussian  beam, all the pulse energy in the Bessel beam impinges  with a relatively large incidence angle toward the plasma  rod generated early in the pulse along the optical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  This configuration is ideal to trigger resonance absorp-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' In reference [16], we have demonstrated the occur-  rence of resonance absorption inside sapphire by com-  paring experimental results to particle-in-cell (PIC) sim-  ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The strong energy transfer between the laser  wave to the core of the plasma opens the possibility to  reach the warm dense matter regime and to produce  temperatures on the order of 10 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' This explains the  opening of high aspect ratio nano channels inside several  dielectrics upon Bessel beam femtosecond illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  Importantly, the conical geometry of Bessel beams is in-  variant along the propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' This implies that results  obtained with Bessel beams of only several tens of mi-  crometers in length can be extrapolated to several cen-  timeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  A key question is the understanding of the micro-  physics of the interaction of the Bessel beam with the  sub-micron plasma, at moderate intensities (typically  1014 W/cm2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  For this, we have used Particle-In-Cell  (PIC) simulations to investigate the electron plasma wave  generation, particle heating and acceleration, as well as  the radiation emitted by the accelerated particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  SIMULATIONS  Our PIC simulations are based on the interaction of a  Bessel-Gauss beam [19] with a preformed plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The  100 fs laser pulse, with a central wavelength of 800 nm,  is polarized along x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We assume that nonlinear  ionization has produced early in the pulse a plasma and  we model the interaction of the most intense part of the  laser pulse with this pre-plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We used the numerical  code EPOCH [20], without ionization, and the numerical  scheme is detailed in reference [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We maintained the  numerical heating at a negligible level over the duration  of the simulation (320 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The pre-plasma is a plasma  rod extending over the whole longitudinal length of the  box (because of the invariance of the Bessel beam), and  its transverse cross-section is elliptically-shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  The  profile is Gaussian and is the same as the one match-  ing our experimental results, as shown in reference [16]:  the critical radius is 190 nm along the polarization di-  rection (x axis) and 380 nm in the other direction, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The peak intensity of the pulse is  6×1014W/cm2, for a pulse duration of 100 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  FIELD AMPLIFICATION AND PARTICLE  ACCELERATION  Figure 1(b) shows the evolution of the Ex component  of the electric field in time, on a segment placed along the  x direction at the propagation distance corresponding to  the highest intensity reached in the Bessel-Gauss beam  [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We first observe the field enhancement in the region  where the permittivity decreases because of the presence  of plasma ( |x| < 190 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' A strong field amplification  occurs on the critical surfaces, that are indicated with  dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The field reaches a maximum  value of 1 GV/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' This corresponds to an amplification  factor of approximately 7 in comparison with the input  laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  At the critical surface, the resonance absorption takes  place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  The wave conversion phenomenon creates elec-  tron plasma waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' 2(a) the density of  electrons at the peak of the pulse, in x − z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The  white line shows the contour at the critical density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We  see plasma oscillations taking place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' To allow a clearer  visualization of the plasma waves, we computed the elec-  trostatic field, that is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' It has been de-  rived from the density ρ using Gauss’s law in the Fourier  k-space, EES  x  = −jkρ(k)/k2/ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  We can observe that, in the sub- critical region, out-  side the plasma, the electron plasma waves are quickly  damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' This arises from the efficient Landau damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  In the over-critical region, the field oscillations are due to  electron sound waves [21, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' They penetrate relatively  deeply into the overcritical region and we observe that  they are curved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The propagation of these waves into the  overcritical region is attributed to the fact that the tem-  perature is highly inhomogeneous in the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' This  is also the reason of the variation of the spatial period  along x, hence the variation of the apparent curvature of  the fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' This structure will have a strong impact on  the acceleration of electrons as we will see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  The heating of the particles has been investigated in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' In summary, the wave particle energy exchange  is very efficient: we observe the heating of the electron  population mainly around the critical surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  2(c), we show the particle distribution in the phase space  x − px after the pulse (135 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We evaluated the main  component temperature to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='3 eV, and a hot electrons  component at 70 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' In addition, we observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' 2(c),  two tails expanding outwards, at high momentum, out-  side the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  These tails correspond to highly accelerated particles  with energies up to 7 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Figure 1 provides a glimpse on  the physical mechanism of the acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We have se-  lected out 1000 of the most energetic particles and have  traced their trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  1(b), we have plot-  ted a selection of 8 representative trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Two of  them show highly accelerated particles that escape from  the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  The other 6 are accelerated by the same  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' (a) Cross section of the 2D density profile of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The black solid line shows the density profile at y = 0  along x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The critical density nc is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='7 × 1021 cm−3 (b) Ex component of the electric field as a function of time and x  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The solid colored lines correspond to the projection in the plane of 8 particle trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Their energy is indicated  using the color code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' (c-h) Zoom-in views of the acceleration of the electrons as they leave the plasma, together with the exact  value of the field Ex(x, y = yp, z = zp, t) sampled at the particle position (xp,yp,zp, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  (a) Plasma density at the time corresponding to the peak intensity in the plane y = 0, which is parallel to the  polarization direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The white solid line shows the contour at the critical density(b) Longitudinal component of the electric  field, which allows the visualization of the plasma density waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' (c) x − px phase space of the electron population at a time  135 fs after the peak of the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  4  (c)  (d)  (e)  [keV]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='7 fs  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='7 fs  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='7 fs  wu  1 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  n[nc]  0  2  4  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='5  (a)  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='2  Ex[GV/cm]  x[μm]  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='5  1  0  1  100  50  0  50  1  100  150  y[μm]  t - tc [fs]  5  (f)  (a)  (h)  4  2  n  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='7 fs  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='7 fs  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='7 fs  00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='5  5  100  20  (e)  (c)  [μm]  n[nc]  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content="0  50  15  2  +  [xd  [keV/c]  1  1'XIN 601  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='5  0  0  10  10  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='5  (b)  α  [μm]  50  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='5  100  0  15  10  15  5  z[μm]  x[μm]4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  (left)The total energy radiated per unit solid an-  gle per unit frequency from the accelerated electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' (right)  Configuration of the angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The laser polarization is along  the x axis  mechanism, but remain trapped by a static electric field  generated by a double layer formation that will be ex-  plained later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The acceleration mechanism is the same  in all cases: in the different sub-figures 1(c-h), we see  that the particles undergo transit acceleration, which oc-  curs when a particle travels through a highly non-uniform  electromagnetic field [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The electrons gain energy  by riding on the electron plasma wave, that is curved  in the x − t space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The effective acceleration occurs on  a distance of less than 60 nm, when crossing the high  resonance field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Depending on the exact position of the  particle with respect to the plasma wave, the acceleration  is more or less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We see that the curvature of the  plasma wave is a key to obtain a progressive acceleration  of the particle while it remains on the peak of the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  Particle acceleration and heating transfer a fraction of  the electrons away from the main plasma, as it is also  apparent on the x − px representation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' This  generates a so-called double layer on either sides of the  plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' These double layers are apparent in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' 1 be-  cause they generate a static E-field that is superimposed  to the laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Importantly, this static field has an  amplitude that is as high as the resonance field of the  laser pulse itself (on the order of GV/cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' This static E-  field even remains after the laser pulse has vanished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Its  damping is governed by the collisions inside the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  RADIATION PATTERN  One of the major advantages of PIC codes is the pos-  sibility to access the full information about the parti-  cle dynamics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=', the position and the momentum as  a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' If this information can be retrieved  and stored for a number of particles, it is then feasible  to post-process the radiation associated with a partic-  ular set of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The radiation diagnostic uses the  information from the particle trajectories, position and  momentum over time, and determines the energy being  radiated by an accelerated charged particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  Let us consider a particle at position r0 (t) at time  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  At the same time, we observe the radiated electro-  magnetic fields from the particle at position r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Due to  the finite velocity of light, we observe the particle at an  earlier position r0 (t′) where it was at the retarded time  t′ = t − R (t′) /c, where R (t′) = |r − r0 (t′) | is the dis-  tance from the charged particle (at the retarded time t′ )  to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Using the Li´enard-Wiechert potentials,  the total energy W radiated per unit solid angle per unit  frequency from a charged particle moving with instanta-  neous velocity β = v/c under acceleration ˙β = a/c can  be expressed as [27]:  d2W  dωdΩ ∝  �����  � ∞  −∞  ˆn × [(ˆn − β) × ˙β]  (1 − β · ˆn)2  eiω(t−ˆn·r(t)/c)dt  �����  2  (1)  Here, n = R (t′) /|R (t′) | is a unit vector that points  from the particle retarded position towards the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  The observer’s viewing angle is set by the choice of  n (ˆx sin θ cos φ + ˆy sin θ sin φ + ˆz cos θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  Figure 3 shows the total radiated energy from an en-  semble of 100 randomly selected electrons traced in the  simulations that we computed using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The distri-  bution is plotted as a function of the angles (θ, φ) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  3(left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The forward direction corresponds to values of  θ below 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We observe that the forward signal shows  maxima at (θ, φ) ≈ (0, π/2) and (0, 3π/2), perpendicu-  lar to the electron acceleration in the x−direction, which  indeed corresponds to the pump laser polarization direc-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The emission follows the well-known power distri-  bution per solid angle Ω emitted by a single particle [27]:  dP  dΩ ∝  | ˙β|2  (1 − β cos θ)3  �  1 −  sin2 θ cos2 φ  γ2(1 − β cos θ)2  �  (2)  where γ is the Lorentz factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  Noticeably, the power shows a much higher signal in  the angles corresponding to the forward direction than for  the backward direction (90◦ < θ ≤ 180◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' This behaviour  could be explained by the coherence of the phases of the  dipole moments induced along the plasma rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' A more  detailed analysis of the radiation spectrum, out of the  scope of the present article, shows the presence of second  harmonic generation and of THz radiation in the forward  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  In conclusion, we have investigated the interaction be-  tween a moderately intense laser pulse shaped as a Bessel  beam with a nanoplasma rod using particle-in-cell sim-  ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We have demonstrated that resonance absorp-  tion generate plasma waves inside plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' These plasma  dW/dw/dQ[a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' u]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='7e-07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='5e-06  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='7e-06  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='8e-06  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='0e-06  180  Z1  150  120  Y  90  60  30-  0  0  60  120  180  240  300  360  [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=']Φ5  waves are highly damped in the sub- critical region while  plasma sound waves can propagate over several hundreds  of nanometers inside the overcritical plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The anal-  ysis of the trajectories of the most energetic particles  shows that the main acceleration mechanism is transit  acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' It occurs at the critical layer when parti-  cles are trapped inside a plasma wave and gain energy  until they are released at the critical surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Because of  the heating of the electron gas, two double layers form on  either sides of the nano-plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' The most energetic par-  ticles, with energies up to 7 keV can escape the plasma  while part of the hot electrons remain trapped by the  potential well due to the electrostatic field of the dou-  ble layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Overall, our results enable us to gain insights  into the micro physics of the laser-plasma interaction  that this relevant for the understanding of the different  mechanisms of the deposition of the femtosecond laser  pulse energy inside dielectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Our results reveal a rich  physics which can be exploited in several fields of appli-  cations: laser-matter interaction, laser micro-machining,  warm dense matter and high energy density physics in-  side solids, as well as the generation of electrostatic fields  and terahertz radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  Acknowledgments :  The research leading to these  results has received funding from the European Re-  search Council (ERC) under the European Union’s Hori-  zon 2020 research and innovation program (grant agree-  ment No 682032-PULSAR), R´egion Bourgogne Franche-  Comt´e, I-SITE BFC project (contract ANR-15-IDEX-  0003), and the EIPHI Graduate School (ANR-17-EURE-  0002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' We acknowledge the support of PRACE HPC re-  sources under the Project ”PULSARPIC” (PRA19 4980  and RA5614), and GENCI resources under projects  A0070511001 and A0090511001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  Data availability statement: Data will be made avail-  able on reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  ∗ kazem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='arrdaneh@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='com  † francois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='courvoisier@femto-st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='fr  [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Gattass and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Mazur, Nature Photonics 2, 219  (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
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+page_content=' Anisi-  mov, Journal of Physics D: Applied Physics 50, 193001  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
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+page_content=' Boing, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' Cav-  alleri,  and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content=' von der Linde, Physical Review B 58,  R11805 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
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+page_content=' Lopez,  in Frontiers in Ultrafast Optics: Biomedical, Scientific,  and Industrial Applications XVI, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE0T4oBgHgl3EQffwBQ/content/2301.02408v1.pdf'}
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+Learning Vision-based Robotic Manipulation Tasks Sequentially in
+Offline Reinforcement Learning Settings
+Sudhir Pratap Yadav1, Rajendra Nagar2, and Suril V. Shah3
+Abstract— With the rise of deep reinforcement learning (RL)
+methods, many complex robotic manipulation tasks are being
+solved. However, harnessing the full power of deep learning re-
+quires large datasets. Online-RL does not suit itself readily into
+this paradigm due to costly and time-taking agent environment
+interaction. Therefore recently, many offline-RL algorithms
+have been proposed to learn robotic tasks. But mainly, all
+such methods focus on a single task or multi-task learning,
+which requires retraining every time we need to learn a new
+task. Continuously learning tasks without forgetting previous
+knowledge combined with the power of offline deep-RL would
+allow us to scale the number of tasks by keep adding them
+one-after-another. In this paper, we investigate the effectiveness
+of regularisation-based methods like synaptic intelligence for
+sequentially learning image-based robotic manipulation tasks
+in an offline-RL setup. We evaluate the performance of this
+combined framework against common challenges of sequential
+learning: catastrophic forgetting and forward knowledge trans-
+fer. We performed experiments with different task combinations
+to analyze the effect of task ordering. We also investigated the
+effect of the number of object configurations and density of
+robot trajectories. We found that learning tasks sequentially
+helps in the propagation of knowledge from previous tasks,
+thereby reducing the time required to learn a new task.
+Regularisation based approaches for continuous learning like
+the synaptic intelligence method although helps in mitigating
+catastrophic forgetting but has shown only limited transfer of
+knowledge from previous tasks.
+I. INTRODUCTION
+Robots now have the capability to learn many single
+manipulation tasks using deep Reinforcement Learning (RL),
+such as pick-place [1], peg-in-hole [23], Cloth folding [14],
+and tying rope knots [17]. Multitask RL has also been applied
+successfully to learn robotic manipulation tasks [8], [10].
+Number of tasks and task-data distribution are kept fixed in
+the case of multi-task RL. Therefore, agent has to be trained
+from scratch whenever it needs to learn a new task, even
+if there is a substantial overlap between tasks. Scaling this
+approach to learn all manipulation tasks at par with humans
+is not feasible. Humans use the experience of previous tasks
+for learning a new task and do not need to learn from the
+start. The sequential (or continual) learning approach tries
+to address this problem by providing a framework where an
+agent can learn new tasks one-after-another without starting
+from scratch. We use offline-RL as the base framework to
+learn a single image-based robotic-manipulation task and
+*This work was done with collaboration of IIT Jodhpur and iHub-Drishti
+Foundation, IIT Jodhpur
+1Sudhir
+Pratap
+Yadav,
+iHub
+Drishti
+Foundation
+[email protected]
+2Rajendra Nagar, IIT Jodhpur [email protected]
+3Suril V. Shah, IIT Jodhpur [email protected]
+Fig. 1: Block Diagram of SAC-CQL-SI method for Sequen-
+tial Learning
+then use a regularisation based continual learning approach
+for learning tasks sequentially. This combined framework
+forms main contribution of this work.
+A. Related Work
+Most work in sequential task learning is focused on
+classification-based tasks using typical classification datasets
+such as MNIST, CIFAR and their variations [7], [15]. Some
+works in continual reinforcement Learning setup use Atari-
+games [12]. Other try to extend continual RL to GYM
+environments [2]. Recent work in [20] uses Offline-RL for
+solving manipulation tasks using image observations alone.
+While this work focuses on generalizing to novel initial
+conditions but does not attempt sequential task learning. On
+the other hand, our work is about learning tasks sequentially
+with only the current task data available for learning. Some
+very recent works try to apply continual RL on robotics
+manipulation tasks [22], [4]. [22] introduces a continual
+learning benchmark for robotic manipulation tasks. It gives
+baselines for major continual learning methods over these
+robotics tasks in online RL settings using soft actor-critic
+(SAC) method [9]. But this work focuses on online-continual
+RL with low-dimensional observation space such as joint and
+task space data as they assume full access to the simulator.
+While our work focuses on Offline RL with high-dimensional
+observation space (image) in sequential learning of robotics
+manipulation tasks.
+In the sequential learning setup based on deep RL, neural-
+networks (NN) are prone to change in data-distribution.
+Hence, its accuracy on previous tasks drops significantly
+when it is trained on a new task. This problem is actively
+studied under the name of catastrophic forgetting. More
+broadly, in every connectionist model of memory and com-
+putation problem of stability-plasticity exists. This means
+arXiv:2301.13450v1  [cs.RO]  31 Jan 2023
+
+Sequential
+Update
+Task Data
+training
+NNs for
+Yes
+Weights
+Actor and
+step > △
+Critic
+Task
+No
+'s"e"s>
+Index (k)
+Add SI loss
+.
+Q value,
+training
+to Actor
+Policy
+No
+Loss
+step ++
+action
+TaskData
+(Tk)
+Yes
+0(1)
+task
+Batch
+Mini
+SAC-CQL
+Actor, Critic
+index
+Sampler
+batch
+Algo
+.. .
+Loss
+(w)othe network needs to be flexible enough to accommodate
+new information and simultaneously not forget previous
+information, as discussed in [6]. Many solutions have been
+suggested to mitigate this problem. We place these under
+two major categories architectural and penalty based. In the
+architectural type of solutions, relevant changes are made in
+the neural network architecture without changing the loss
+function. For example, Progressive neural network [19] uses
+multiple parrallel paths with lateral connenctions, and policy
+distillation [18] distills the policy learned by larger network
+into smaller without loss of performance. On the other
+hand, penalty based methods put penalties on neural network
+parameters so that they stay close to solution of previous
+task. Two important work in this regards are Elastic Weight
+Consolidation (EWC) [12] and Synaptic Intelligence [24].
+EWC gives regularisation based solution for catastrophic
+forgetting, but computation for the importance of parameters
+is not local. In this paper we use the approach proposed
+in Synaptic Intelligence [24] because of its local measure
+of importance for synapses (weights of Neural Network)
+as the nature of local computation helps in keeping the
+solution independent of the particularities of the problem
+hence making it more general.
+To the best of our knowledge, this is the first work
+investigating sequential learning for image based robotic
+task manipulation in offline RL settings. In this paper, we
+focus on two sequential learning challenges: catastrophic
+forgetting and forward knowledge transfer. We analyse the
+effect of task-ordering and number of object configurations
+on forgetting and forward knowledge transfer between tasks.
+II. LEARNING IMAGE BASED ROBOTIC
+MANIPULATION TASKS SEQUENTIALLY
+In this section, we formulate our RL agent and environ-
+ment interaction setup to learn robotic manipulation tasks.
+We then discuss the problem of sequential task learning and
+present an approach to solve this problem.
+A. RL formulation for Learning Image Based Robotic Ma-
+nipulation Tasks
+Agent and environment interaction is formally defined by
+the Markov Decision Process (MDP) concept. A Markov De-
+cision Process is a discrete-time stochastic control process.
+In RL, we formally define the MDP as a tuple ⟨S, A, P, r, γ⟩.
+Here, S is a finite set of states, A is a finite set of actions,
+P is the state transition probability matrix, r is the reward
+for a given state-action pair and γ is the discount factor.
+A stochastic policy is defined as a distribution over actions
+given the states, i.e., the probability of taking each action for
+every state. π(a|s) = P[At = a|St = s].
+RL formulation: We formulate the vision-based robotic
+manipulation tasks using the deep RL framework as below.
+• Environment: It consists of WidowX 250 five-axes
+robot arm equipped with a gripper. We place a table
+in front of the robot. Every task consists of an object
+placed on the table, which needs to be manipulated to
+complete the task successfully. We place a camera in
+the environment in eye-to-hand configuration.
+• State: The state st represents the RGB image of the
+environment captured at time step t. We use capture
+images of size 48 × 48 × 3.
+• Action: We define the action at the time step t as a
+7 dimensional vector at =
+�∆Xt
+∆Ot
+gt
+�⊤. Here,
+∆Xt ∈ R3, ∆Ot ∈ R3, gt ∈ {0, 1} denotes the change
+in position, change in orientation, and gripper command
+(open/close), respectively at time step t.
+• Reward: The reward r(st, at) ∈ {0, 1} is a binary
+variable which is equal to 1 if the task is successful
+and 0, otherwise. Reward is given at each time step.
+The reward is kept simple and not shaped according to the
+tasks so that the same reward framework can be used while
+scaling for large number of tasks. Also, giving reward at
+each time step, instead at the end of the episode, makes the
+sum of rewards during an episode dependent on time steps.
+Therefore, if the agent completes a task in fewer steps, its
+total reward for that episode will be more.
+B. Sequential Learning Problem and Solution
+We define the sequential tasks learning problem as follows.
+The agent is required to learn N number of tasks but with
+the condition that tasks will be given sequentially to the
+agent and not simultaneously. Therefore, when the agent is
+learning to perform a particular task, it can only access the
+data of the current task. This learning process reassembles
+how a human learns. Let a sequence of robotic manipulation
+tasks T1, T2, ..., TN is given. We assume that each task
+has the same type of state and action space. Each task
+has its own data in typical offline reinforcement learning
+format ⟨st, at, rt, st+1⟩. The agent has to learn a policy
+π, a mapping from state to action, for every task. If we
+naively train a neural network in this fashion problem of
+catastrophic forgetting will occur, which means performance
+on the previous task will decrease drastically as soon as the
+neural network starts learning a new task.
+We use a regularisation based approach presented in [24]
+to mitigate the problem of catastrophic forgetting. Figure 1
+provides the overall framework we use to solve this problem.
+Each task data is given one by one to the algorithm, which
+then starts training for the current task. First, a mini-batch
+is sampled from this current-task data and passed to the
+SAC-CQL algorithm (described in the next section), which
+then calculates actor (Q-loss) and critic (policy) loss. If
+the task-index is greater than one then we add a quadratic
+regularisation as defined in [24] to the actor loss to reduce
+forgetting. Then, these losses are used to update neural
+networks, which represents policy (actor network) and Q-
+value function (critic network). After the current task is
+successfully learned next task data comes, and this process
+is repeated until all tasks are learned.
+
+III. INTEGRATING SEQUENTIAL TASK
+LEARNING WITH OFFLINE RL
+In this section, we discuss the SAC-CQL [13] offline
+algorithm and its implementation details. We then discuss the
+SI regularisation method for continual learning and provide
+details to integrate these methods to learn sequential tasks.
+A. SAC-CQL algorithm for Offline RL
+There are two frameworks, namely online and offline
+learning, to train an RL agent. In the case of an online-
+RL training framework, an RL agent interacts with the envi-
+ronment to collect experience, update itself (train), interact
+again, and so on. In simple terms, the environment is always
+available for the RL agent to evaluate itself and improve
+further. This interaction loop is repeated for many episodes
+during training until the RL agent gets good enough to
+perform the task successfully. While in offline RL settings,
+we collect data once and are no more required to interact with
+the environment. This data can be collected by executing a
+hand-designed policy or can be obtained by a human con-
+trolling the robot (human-demonstration). Data is a sequence
+of ⟨st, at, rt, st+1⟩ tuples.
+In recent years the SAC (soft-actor critic) [9] has emerged
+as the most robust algorithm for training RL agents in
+continuous action space (when action is a real vector), which
+typically is a case in robotics. SAC is an off-policy entropy
+based actor-critic method for continuous action MDPs. En-
+tropy based methods add an additional entropy term to the
+existing optimisation goal of maximising expected reward. In
+addition to maximising expected reward, the RL agent also
+needs to maximise the entropy of the overall policy. This
+helps in making the policy inherently exploratory and not
+stuck inside a local minima. Haarnoja et al. [9] define the
+RL objective in maximum entropy RL settings as in (1).
+J(π) =
+T
+�
+t=0
+E(st,at)∼ρπ[r(st, at) + αH(π(·|st))].
+(1)
+Here, ρπ(st, at) denotes the joint distribution of the state
+and actions over all trajectories of the agent could take and
+H(π(·|st)) is the entropy of the policy for state st as defined
+in (2).
+H(π(·|st)) = E[−log(fπ(·|st))].
+(2)
+Here, π(·|st) is a probability distribution over actions and
+fπ(·|st) is the density function of the policy π. α is the
+temperature parameter controlling the entropy in the policy.
+SAC provides an actor-critic framework where policy is
+separately represented by the actor and critic only helps in
+improving the actor, thus limiting its role only to training.
+We use CNNs to represent both actor and critic, and instead
+of using a single Q-value network for the critic, we use two
+Q-value networks and take their minimum to better estimate
+Q-value as proposed in [21]. To stabilize the learning, we
+use two more neural-network to represent target Q-values
+for each critic network, as described in DQN [16]. φ, θ1,
+θ2, ˆθ1 and ˆθ2 represents parameters of policy network, 2
+Q-value networks and 2 target Q-value networks for critic
+respectively. Therefore in total, we use 5 CNNs to implement
+the SAC algorithm.
+Our CNN architecture is similar to [20] except for the
+multi-head part, which is a single layer neural-network for
+each head. Q-value network takes state and action as input
+and directly gives Q-value. We use tanh-guassian policy, as
+used in [20]. Since we use stochastic policy thus, the policy
+network takes the state as input and outputs the mean and
+standard deviation of the gaussian distribution of each action.
+Action is then sampled from this distribution and passed
+through tanh function to bound actions between (−1, 1).
+Equation (3) defines the target Q-value which is then used
+in (4) to calculate Q-loss for each critic networks. Equation
+(5) defines policy-loss for actor network. These losses are
+then used to update actor and critic networks using adam
+[11] optimisation algorithm.
+ˆQ¯θ1,¯θ2(st+1, at+1) = rt
++ γE(st+1∼D,at+1∼πφ(·|st+1))[
+min[Q¯θ1(st+1, at+1), Q¯θ2(st+1, at+1)]
+− αlog(πφ(at+1|st+1))]
+(3)
+JQ(θi) = 1
+2E(st,at)∼D[( ˆQ¯θi,¯θ2(st+1, at+1) − Qθ1(st, at))2].
+(4)
+Jπ(φ) = E(st∼D,at∼πφ(·|st))[αlog(πφ(at|st))
+− min[Qθ1(st, aπ
+t ), Qθ2(st, aπ
+t )]]
+(5)
+Here, i ∈ {1, 2}, aπ
+t is the action sampled from policy
+πφ for state st and D represents the current task data.
+For offline-RL, we use the non-Lagrange version of the
+conservative Q-learning (CQL) approach proposed in [13] as
+it only requires adding a regularisation loss to already well-
+established continuous RL methods like Soft-Actor Critic.
+This loss function is defined in (6).
+Jtotal
+Q (θi) = JQ(θi)
++αcqlEst∼D[log
+�
+at
+exp(Qθi(st, at))−Eat∼D[Qθi(st, at)]]
+(6)
+Here, i ∈ {1, 2}, αcql control the amount of CQL-loss to be
+added to Q-loss to penalize actions that are too far away from
+the existing trajectories, thus keeping the policy conservative
+in the sense of exploration.
+B. Applying Synaptic Intelligence in Offline RL
+Synaptic intelligence is a regularisation based algorithm
+proposed in [24] for sequential task learning. It regularises
+the loss function of a task with a quadratic loss function as
+defined in (7) to reduce catastrophic forgetting.
+Lµ =
+�
+k
+Ωµ
+k(˜φk − φk)2
+(7)
+Here, Lµ is the SI loss for the current task being learned
+with index µ, φk is k-th weight of the policy network, and
+
+˜φk is the reference weight corresponding to policy network
+parameters at the end of the previous task. Ωµ
+k is per-
+parameter regularisation strength for more details on how
+to calculate Ωµ
+k refer to [24]. SI algorithm penalizes neural
+network weights based on their contributions to the change
+in the overall loss function. Weights that contributed more
+to the previous tasks are penalized more and thus do not
+deviate much from their original values, while other weights
+help learn new tasks. SI defines importance of weights as
+the sum of the gradients over the training trajectory, as this
+approximates contribution to the reduction in the overall
+loss function. We use a similar approach to apply SI to
+Offline-RL as presented in [22]. Although the authors didn’t
+use SI or offline-RL, the approach is similar to applying
+any regularisation based continual learning method for the
+actor-critic RL framework. We regularise the actor to reduce
+forgetting on previous tasks while learning new tasks using
+offline reinforcement learning. We add quadratic loss as
+defined in [24] to the policy-loss term in the SAC-CQL
+algorithm. So now over-all policy-loss becomes as described
+in (8)
+Jtotal
+π
+(φ) = Jπ(φ) + cLµ
+(8)
+Here, c is regularisation strength. Another aspect of continual
+learning is finding a way to provide the current task index
+to the neural network. There are many approaches to tackle
+this problem, from 1-hot encoding to recognizing the task
+from context. We chose the most straightforward option of a
+multi-head neural network. Each head of the neural network
+represents a separate task. Therefore we simply select the
+head for a given task. For training each task we keep a fixed
+compute budget of 100k of gradient-steps.
+IV. EXPERIMENTS, RESULTS AND DISCUSSION
+In this section we first discuss the RL environment setup
+and provide details of data collection for offline RL. Further,
+we evaluate performance of SI with varying number of object
+configurations and densities for different task ordering.
+A. Experimental Setup
+Our experimental setup is based on a simulated envi-
+ronment, Roboverse, used in [20]. It is a GYM [3] like
+environment based upon open-source physics simulator py-
+bullet [5]. We collected data for three tasks using this
+simulated environment.
+Object Space and Tasks: We define object space as
+a subset of the workspace of the robot where the target
+object of the task is to be placed. In our case, it is a
+rectangular area on the table in front of the robot. The
+target object is randomly placed in the object-space when
+initializing the task. We selected the following 3 tasks for
+all our experiments with some similarities.
+1) Press Button: Button is placed in the object-space. The
+objective of the task is to press the button. This is easiest
+task as the robot only needs to learn to reach the object.
+2) Pick Shed: The objective of this task is to pick the
+object successfully. Thus, robot also needs to learn to close
+(a)
+(b)
+Fig. 2: Object space and reward distribution for pick-shed
+task with area of size of 1000 cm2 and density of 20
+object configurations per cm2. (a) Object Space. (b) Reward
+distribution (cumulative reward along sample trajectories) of
+pick-shed task (area=1000, density=20). Red semicircle on
+top represents the robot location
+the gripper apart from reaching the object. Figure 2(a) shows
+the object space of task pick-shed.
+3) Open Drawer: The objective of this task is to open the
+drawer.
+Data Collection: For each task we collect 6 datasets
+by varying the area (40, 360 and 1000 cm2) and density
+(10 and 20 object configurations per cm2) of object-space.
+Each episode consists of 20 steps and each step is a typical
+tuple < st, at, rt, st+1 > used in reinforcement learning. We
+use simple but accurate policies to collect data. Accuracy
+of these data collection policies is above 80%. Figure 2(b)
+shows how reward is distributed across object space for pick-
+shed task. Each dot represents a trajectory, and the color
+represents the total reward for each trajectory. It can be seen,
+when the object is placed closer to the robot, the reward is
+high as task is completed in few steps, while it becomes low
+as the object moves away.
+B. Empirical Results and Analysis
+We performed a total of 72 experiments. We performed
+sequential learning on two task (doublets). Six doublets
+are possible using data collected for three tasks. These
+are button-shed, button-drawer, shed-button, shed-drawer,
+drawer-shed, and drawer-button. For each doublet sequence,
+we perform 2 sets of experiments, one with SI regularisation
+and another without SI regularisation. Each set contains 6
+experiments by varying area and density of object-space.
+Apart from these 72 experiments, we also trained the agent
+for single tasks using SAC-CQL for reference baseline
+performance to evaluate forward transfer. We do behaviour-
+cloning for the initial 5k steps to learn faster as we have
+limited compute budget. We use metrics mentioned in [22]
+for evaluating the performance of a continual learning agent.
+Each task is trained for ∆ = 100K steps. The total number
+of tasks in a sequence is N = 2. Total steps T = 2 · ∆. The
+i-th task is train from t ∈ [(i − 1) · ∆, i · ∆].
+Task Accuracy: We evaluate the agent after every 1000
+training steps by sampling 10 trajectories from the environ-
+ment for each task. The accuracy of the agent for a task
+is defined as the number of successful trajectories out of
+those 10 trails. Figure 3 shows the accuracy of three task-
+
+X (m)
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+12
+0.0
+10
+0.1
+8
+0.2
+6
+>
+4
+0.3
+2
+0.4
+0(a) Task Accuracy (area=360, density=10)
+(b) Task Accuracy (area=360, density=20)
+Fig. 3: Task accuracy for tasks button-shed, button-drawer and drawer-button (area=360, density=10,20). Top row is with
+SI, bottom row is without SI
+sequences (button-shed, button-drawer, drawer-button) over
+the complete training period of 200k steps for the area size
+of 40cm2 with density of 10 and 20 object configurations per
+cm2. Top row represents sequential learning with SI while
+bottom row represents sequential learning without SI. SI is
+found to be working better as evident by overlapping Task-1
+and Task-2 accuracy. We observed that SI was most helpful in
+button-shed task doublet due to overlapping nature of these
+tasks as both these tasks require reaching the object. This
+shows benefit of using SI for overlapping tasks
+Forgetting: It measures decrease in accuracy of the task
+as we train more tasks and defined as Fi := pi(i.∆)−pi(T).
+Here, pi(t) ∈ [0, 1] is success rate of task i at time t. Figure
+4 shows the forgetting of Task-1 after training Task-2. We
+can see that SI performed better or equal in all cases. In fact,
+in some cases, like button-shed forgetting is negative, which
+means the performance of Task-1 improved after training on
+Task-2. This indicates knowledge transfer from Task-1 to
+Task-2. This phenomenon is not seen in case of sequential
+learning without SI. This clearly indicates that SI helps in
+reducing catastrophic forgetting. No significant trends are
+observed in variation of object-space area but forgetting
+increased with the increase in object-space density. This
+might be due to the limited compute budget (100K) per task
+as tasks with more area size and density would require more
+training to show good results.
+Forward Transfer: It measures knowledge transfer by
+comparing the performance of a given task when trained
+individually versus learning the task after the network is
+already trained on previous tasks and defined as
+FTi := AUCi − AUCb
+i
+1 − AUCb
+i
+,
+(9)
+where AUCi =
+1
+∆
+� i·∆
+(i−1)·∆ pi(t)dt represents area under
+the accuracy curve of task i and AUCb
+i =
+1
+∆
+� ∆
+0 pb
+i(t)dt,
+represents area under curve of the reference baseline task.
+pb
+i(t) represent reference baseline performance. Figure 5
+shows forward transfer for Task-2 after it is trained on Task-
+1. We use single-task training performance as the reference
+for Task-2 while evaluating forward transfer. We observed
+that in most cases, training without SI gives a better transfer
+ratio than training with SI. This may be because of two
+reasons. Firstly, due to the high value of SI regularisation
+strength (which is set to 1 for all cases), this restricts
+movement of weights from the solution of the previous task.
+This can also be noticed in the form of reduced accuracy
+levels of Task-2 in the Figure 3. The accuracy level of Task-2
+are lower as compared to its non-SI counterpart. Although,
+high regularisation strength helps in reducing catastrophic
+forgetting but also hinders the ability to learn new-task thus
+reducing forward-transfer. This highlights the problem of
+stability-plasticity, any method which tries to make learning
+more stable to reduce forgetting inadvertently also restricts
+the flexibility of the connectionist model to learn a new task.
+Training Time: Apart from these metrics, we observed
+that, agent requires on an average 14k, 10k, and 16k steps
+to achieve its first success on Task-2 when trained directly,
+sequentially without SI, and sequentially with SI, respec-
+tively. This means that the agent learns the task faster when
+trained sequentially without adding SI regularisation but a
+little slower when trained sequentially with SI regularisation
+than directly training the task. This shows another benefit of
+sequential learning over single task-learning.
+Another interesting observation we made in the case of
+shed-button (area 360, density 20) task. While training for
+Task-1 (pick shed) agent showed some success on Task-
+2 (press button) even before getting any success on Task-
+1 itself. This might be due to the nature of the tasks, as
+
+Task Accuracy with and without Sl (a:360, d:10)
+button shed
+button drawer
+drawer button
+1.0
+1.0
+1.0
+0.8
+0.8
+0.8
+0.6
+0.6
+0.6
+0.4
+0.4
+0.4
+0.2
+0.2
+0.2
+0.0
+0.0
+0.0
+0.0
+0.51.01.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.51.0
+1.5
+2.0
+Steps
+1e5
+Steps
+1e5
+Steps
+1e5
+button shed
+button drawer
+drawer button
+1.0
+1.0
+1.0
+0.8
+0.8
+0.8
+0.6
+0.6
+0.6
+0.4
+0.4
+0.4
+0.2
+0.2
+0.2
+0.0
+0.0
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.01.52.0
+0.0
+0.5
+1.0
+¥1.52.0
+Steps
+1e5
+Steps
+1e5
+Steps
+1e5
+task 1
+data collection poliy accuracy (task 1)
+task 2
+data collection poliy accuracy (task 2)Task Accuracy with and without Sl (a:360, d:20)
+button shed
+button drawer
+drawer button
+1.0
+1.0
+1.0
+0.8
+0.8
+0.8
+0.6
+0.6
+0.6
+0.4
+0.4
+0.4
+0.2
+0.2
+0.2
+0.0
+0.0
+0.0
+0.0
+0.51.0
+1.5
+2.0
+0.0
+0.5
+1.01.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+Steps
+1e5
+Steps
+1e5
+Steps
+1e5
+button shed
+button drawer
+drawer button
+1.0
+1.0
+1.0
+0.8
+0.8
+0.8
+0.6
+0.6
+0.6
+0.4
+0.4
+0.4
+0.2
+0.2
+0.2
+0.0
+0.0
+0.0
+0.0
+0.5
+1.0
+1.5
+¥2.0
+0.0
+0.5
+1.0
+2.0
+0.0
+0.5
+1.01.52.0
+1.5
+Steps
+1e5
+Steps
+1e5
+Steps
+1e5
+task 1
+data collection poliy accuracy (task 1)
+task 2
+data collection poliy accuracy (task 2)Fig. 4: Forgetting Matrix. Top row is with SI regularisation, bottom row is without regularisation
+Fig. 5: Forward Transfer Matrix
+the trajectory of the press button task is common for other
+task. Therefore, agent has tendency to acquire knowledge
+for similar tasks. This may also be the result of behaviour-
+cloning for the initial 5k steps, where the agent tries to mimic
+the data collection policy for a few initial training steps. Also,
+we observed that increasing the object space area helps in
+knowledge transfer, which can be seen by the increase in
+average forward transfer with area size.
+V. CONCLUSION AND FUTURE WORK
+We
+investigated
+catastrophic
+forgetting
+and
+forward
+knowledge transfer for sequentially learning image-based
+robotic manipulation tasks by combining a continual learning
+approach with offline RL framework. We use SAC-CQL as
+an offline deep RL algorithm with synaptic intelligence (SI)
+to mitigate catastrophic forgetting. Multi-headed CNN was
+used to provide knowledge of the current Task-index to the
+neural-network. We performed a series of experiments with
+different task combinations and with a varying number of
+object configurations and densities. We found that SI is useful
+for reducing forgetting but showed a limited forward transfer
+of knowledge.
+We also found that the ordering of tasks significantly af-
+fects the performance of sequential task learning. Therefore,
+tasks may be chosen in a manner so that the previous task
+helps in learning the next task as the complexity of tasks
+increases. This calls for exploring curriculum learning for
+sequential tasks. Experiments also suggests the importance
+of prior knowledge for continual learning. Agent trained
+only with state-action pairs of large number of diverse tasks
+(even without reward), may provide a better prior knowledge.
+Future work will also focus on training tasks with more
+number of steps to explore more interesting patterns.
+
+Forgetting Matrix (O-button, 1-shed, 2-drawer)
+f_avg = 0.2
+f_avg = 0.23
+f_avg = -0.03
+f_avg = 0.4
+f_avg = 0.23
+f_avg = 0.25
+(d:10, a:40)
+(d:10, a:360)
+(d:10, a:1000)
+(d:20, a:40)
+(d:20, a:360)
+(d:20, a:1000)
+-0.6
+0
+0
+-0.4
+0.8
+0
+0.1
+-0.1
+0
+-0.3
+0
+0
+0.4
+0.1
+0
+0.1
+0.2
+0
+0.1
+0.2
+0.4
+0.1
+0
+0.4
+0.2
+0
+0.2
+0
+0
+0.1
+0.1
+0
+0.6
+0.1
+0
+0
+0.1
+0
+0.1
+1
+- 0.2 
+2
+0.2
+0.1
+0
+0.5
+0.5
+0
+0
+0
+0
+0.6
+0.6
+0
+0.5
+0.5
+0
+0.5
+0.5
+0
+1
+1
+0
+2
+0
+2
+0
+1
+2
+2
+0
+1
+0
+0
+2
+1
+2
+- 0.0 
+f_avg = 0.37
+f_avg = 0.23
+f_avg = 0.03
+f_avg = 0.4
+f_avg = 0.25
+f_avg = 0.25
+(d:10, a:40)
+(d:10, a:360)
+(d:10, a:1000)
+(d:20, a:40)
+(d:20, a:360)
+(d:20, a:1000)
+0
+0
+0
+0.5
+0.8
+0
+0.1
+0.1
+0
+0.4
+0.1
+0
+0.2
+0.2
+0
+0
+0.1
+0.2
+0
+-0.2
+task 1
+0.1
+0
+0.4
+0.2
+0
+0.2
+0
+0
+0.1
+0.1
+0
+0.6
+0.1
+0
+0
+0.1
+0
+0.1
+1
+ -0.4
+0.2
+0.2
+0
+0.4
+0.5
+0
+0.6
+0.6
+0
+0
+0
+0.5
+0.5
+0
+2
+0.5
+0.5
+0
+-0.6
+0
+1
+2
+0
+2
+0
+1
+2
+0
+1
+2
+0
+2
+1
+0
+1
+2
+task 2
+task 2
+task 2
+task 2
+task 2
+task 2Forward Transfer Matrix (O-button, 1-shed, 2-drawer)
+ft_avg = -0.22
+ft_avg = -0.13
+ft _avg = -0.11
+ft avg = -0.07
+ft_avg = -0.03
+ft_avg = -0.13
+(d:10, a:40)
+(d:10, a:360)
+(d:10, a:1000)
+(d:20, a:40)
+(d:20, a:360)
+(d:20, a:1000)
+0
+-0.21
+-0.27
+0
+-0.013
+-0.16
+0
+-0.054
+-0.24
+-0.071
+0.15
+0
+-0.0058
+0.096
+0
+0.0094
+-0.32
+0
+- 0.3
+0
+-0.06
+-0.29
+-0.11
+-0.16
+-0.038
+-0.24
+-0.19
+0
+0.17
+-0.026
+-0.059
+-0.035
+0
+-0.34
+0
+0
+0
+0
+1
+ 0.2 
+-0.12
+-0.37
+-0.15
+-0.2
+0
+-0.049
+-0.055
+-0.25
+-0.23
+0
+-0.021
+-0.15
+2
+0
+0
+0
+-0.068
+-0.043
+0
+- 0.1
+1
+0
+2
+0
+2
+0
+1
+2
+0
+2
+0
+1
+2
+0
+1
+2
+- 0.0 
+ft_avg = -0.02
+ft_avg = -0.0
+ft_avg = -0.02
+ft_avg = 0.1
+ft_avg = 0.09
+ft_avg = -0.03
+(d:10, a:40)
+(d:10, a:360)
+(d:10, a:1000)
+(d:20, a:40)
+(d:20, a:360)
+(d:20, a:1000)
+-0.1
+0
+-0.045
+-0.1
+0
+0.06
+0.021
+0
+0.045
+0.035
+0
+0.11
+0
+0.042
+0.14
+0
+0.062
+-0.15
+0
+0.22
+task 1
+-0.2
+0.18
+0
+-0.22
+0.0048
+0
+-0.061
+-0.019
+0
+-0.12
+0.11
+0
+0.18
+0.12
+0
+0.18
+0.021
+0
+-0.14
+1
+-0.3
+0.13
+-0.03
+0
+0.025
+-0.064
+0
+-0.043
+0
+0
+-0.06
+0.015
+0
+0.063
+-0.0012
+0
+-0.018
+0.052
+2
+0
+2
+0
+1
+2
+0
+1
+0
+2
+0
+1
+2
+0
+1
+2
+0
+1
+2
+task 2
+task 2
+task 2
+task 2
+task 2
+task 2REFERENCES
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+Tuomas Haarnoja et al. “Soft actor-critic: Off-policy
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+Dmitry Kalashnikov et al. “Scaling Up Multi-Task
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+[11]
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+James Kirkpatrick et al. “Overcoming catastrophic
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+Arun
+Mallya
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+on Computer Vision and Pattern Recognition. 2018,
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+[16]
+Volodymyr
+Mnih
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+al.
+“Playing
+atari
+with
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+Ashvin Nair et al. “Combining self-supervised learn-
+ing and imitation for vision-based rope manipulation”.
+In: 2017 IEEE international conference on robotics
+and automation (ICRA). IEEE. 2017, pp. 2146–2153.
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+Andrei A Rusu et al. “Policy distillation”. In: arXiv
+preprint arXiv:1511.06295 (2015).
+[19]
+Andrei A Rusu et al. “Progressive neural networks”.
+In: arXiv preprint arXiv:1606.04671 (2016).
+[20]
+Avi Singh et al. “Cog: Connecting new skills to past
+experience with offline reinforcement learning”. In:
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+[21]
+Hado Van Hasselt, Arthur Guez, and David Silver.
+“Deep reinforcement learning with double q-learning”.
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+intelligence. Vol. 30. 1. 2016.
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+Maciej Wołczyk et al. “Continual world: A robotic
+benchmark for continual reinforcement learning”. In:
+Advances in Neural Information Processing Systems
+34 (2021), pp. 28496–28510.
+[23]
+Andr´e Yuji Yasutomi, Hiroki Mori, and Tetsuya Ogata.
+“A peg-in-hole task strategy for holes in concrete”. In:
+2021 IEEE International Conference on Robotics and
+Automation (ICRA). IEEE. 2021, pp. 2205–2211.
+[24]
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+In: International Conference on Machine Learning.
+PMLR. 2017, pp. 3987–3995.
+

ANFAT4oBgHgl3EQfrR6g/content/tmp_files/2301.08652v1.pdf.txt

@@ -0,0 +1,1415 @@
+KCL-PH-TH/2023-05
+Phantom-like equation of state from tunnelling
+Jean Alexandre1 and Silvia Pla1
+1Theoretical Particle Physics and Cosmology, King’s College London, WC2R 2LS, UK
+We allow a scalar field on a flat FLRW background metric to tunnel between two degenerate vacua.
+The resulting true vacuum state then violates the Null Energy Condition, and the corresponding
+homogeneous fluid has a phantom-like equation of state. The mechanism presented here requires no
+exotic matter or modified gravity, it is purely generated by quantum fluctuations and is valid for a
+generic double well potential.
+CONTENTS
+I. Introduction
+2
+II. Semi-classical approximation and saddle points
+2
+A. Assumptions
+2
+B. Semi-classical approximation
+3
+C. Static saddle points
+4
+D. Instanton gas
+4
+III. Effective action
+5
+A. Symmetric ground state
+6
+B. NEC violation
+7
+IV. Friedmann Equations
+8
+V. Conclusions
+8
+Acknowledgements
+10
+A. One-loop effective action in curved space-times
+10
+B. Quantisation over instanton configurations
+12
+C. Effective action, energy density and pressure
+13
+References
+15
+arXiv:2301.08652v1  [hep-th]  20 Jan 2023
+
+2
+I.
+INTRODUCTION
+Unlike spontaneous symmetry breaking (SSB), which occurs in infinite volume, tunnelling involves remarkable
+energetic features, among which a non-perturbative ground state with no classical analogue. This is at the origin of
+convexity of the effective potential for a scalar field [1–9], and thus symmetry restoration.
+The explicit calculation of the one-particle-irreducible (1PI) effective potential, taking into account several degener-
+ate vacua, was done in [10, 11] in the semi-classical approximation for the partition function. These studies assumed
+an O(4)-symmetric Euclidean space-time, and the corresponding work at finite but low temperature was done in [12]
+and [13]. The latter works allow for the full tunnelling regime, involving a gas of Euclidean-time-dependent instantons
+relating two degenerate vacua. It was found that the true ground state for the scalar field is symmetric, and it violates
+the Null Energy Condition (NEC - see [14, 15] for reviews), because it is non-extensive in the thermodynamical sense.
+We note here that these results are independent of symmetry-restoration by the Kibble-Zurek mechanism [16, 17],
+which is valid at high temperatures and does not allow for the NEC to be violated.
+The present work extends this tunnelling mechanism to a Friedmann-Lemaitre-Robertson-Walker (FLRW) back-
+ground metric, where we study the backreaction of the fluid provided by the scalar true vacuum on the metric
+dynamics. Our assumptions do not involve exotic matter or modified gravity, but a finite volume and an adiabatic
+expansion instead, both to be defined in the next section. Our results arise purely from quantum fluctuations and
+they have no classical counterpart.
+It is well known that in Quantum Field Theory (QFT) the energy conditions can be violated under certain circum-
+stances. Some examples include the Casimir effect [18], radiation from moving mirrors [19], or black hole evaporation
+[20].
+Another interesting example in curved backgrounds was obtained in [21].
+The latter work studies a self-
+interacting massless field, therefore seeing only one vacuum, and not tunnelling. Also, the background is fixed as a de
+Sitter metric, whereas in our study the scale factor is determined by the backreaction of the scalar effective vacuum.
+Nevertheless, it is still possible for the stress-energy tensor to satisfy certain constraints, such as the Averaged Null
+Energy Condition (ANEC), which averages the NEC over timelike or null geodesics. The mechanism we propose here
+indeed does not violate the ANEC, since NEC violation is valid temporarily only - see Sec.IV. We note that an eternal
+inflation scenario is described in [22], which also respects the ANEC.
+In Sec.II we describe the semi-classical approximation in which we evaluate the partition function, based on the dif-
+ferent saddle points which are relevant for two degenerate vacua: two static configurations and a gas of instantons/anti-
+instantons. In the situation of non-degenerate vacua, the relevant configurations are the Coleman bounce [23, 24] and
+the shot [25], with imaginary quantum fluctuations which arise from a negative eigenvalue in the fluctuation determi-
+nant [26]. In the present case though, there are not any imaginary quantum corrections, since the (anti-)instantons
+are monotonic functions of Euclidean time [27].
+The effective action is then derived in Sec.III to the lowest order in the field, which is enough to confirm convexity
+and that the ground state is obtained for a vanishing field, unlike the situation of SSB. This calculation is done in the
+adiabatic approximation, assuming that the tunnelling rate is large compared to the space-time expansion rate. The
+vacuum energy induced by tunnelling violates the NEC and has an equation of state of the form
+w = −1 − ℏ w0
+�
+α3/2(t) +
+1
+2α3/2(t)
+�
+e−α3(t) ,
+(1)
+where w0 > 0 and α(t) is proportional to the scale factor. The property w < −1 is usually related to a negative
+kinetic term in the potential (see [28] for a review on phantom energy), but is not the case here: the vacuum we find
+is homogeneous and its energetic properties arise purely from quantum fluctuations, not from a specific bare action.
+In Sec.IV we solve numerically the Friedmann equations, where we study the backreaction of the effective theory on
+gravity. As expected from NEC violation, the solution exhibits a cosmological bounce [29–31], known to provide an
+alternative to Cosmic Inflation [32, 33]. The original idea to generate a bounce from a tunnelling-induced scalar field
+true vacuum was proposed in [34, 35], in the context of an O(4)-symmetric Euclidean space-time though, whereas we
+allow here for the full tunnelling regime, with finite volume and infinite Euclidean time.
+Finally, the detailed calculations are presented in Appendix A, B and C.
+II.
+SEMI-CLASSICAL APPROXIMATION AND SADDLE POINTS
+A.
+Assumptions
+We consider the classical background metric
+ds2 = −dt2 + a2(t)δijdxidxj ,
+(2)
+
+3
+where the scale factor a(t) is kept generic. The bare matter action is
+S[φ] =
+�
+d4x
+�
+|g|(L − jφ) ,
+(3)
+where the Lagrangian L involves a double-well potential, as well as a non-minimal coupling to the scalar curvature:
+L = −1
+2gµν∂µφ∂νφ − 1
+2ξRφ2 − λ
+4!(φ2 − v2)2 − ¯Λ .
+(4)
+For convenience, we have also added the cosmological constant term in the matter sector (¯Λ = κ−1Λ with κ = 8πG)
+to account for vacuum energy effects after renormalisation. The important assumptions we make are the following:
+• Finite volume, which allows tunnelling between the degenerate vacua. We start from a fundamental flat spatial
+cell with volume V0 and comoving volume a3(t)V0, which can be thought of as a 3-torus, or a 3-sphere with large
+enough radius to neglect curvature. Although finite, we assume the parameters of the model to be such that
+quantisation of momentum can be ignored, and the periodic boundary conditions do not play a role. Related
+comments on the Casimir effect are given in [13] for tunnelling in flat space-time, and we focus here on the
+tunnelling features only;
+• Adiabatic approximation, where the expansion rate of the metric is assumed small compared to the tunnelling
+rate for matter. According to the discussion at the end of Sec.II D, this is valid in the regime
+|H| ≡
+����
+˙a
+a
+���� ≪ v
+�
+λ
+π α3/2 exp(−α3) ,
+(5)
+where α3(t) = a3(t)S0/ℏ and S0 is the action for an instanton interpolating the two vacua ±v.
+As a consequence of the second point, the scale factor a(t) will be considered constant for the calculation of the matter
+effective theory, and its time dependence will be reinstated when we couple the matter effective theory to gravity.
+B.
+Semi-classical approximation
+We work here in Euclidean signature. In the semi-classical approximation, and focusing only on the matter sector
+for the reasons explained above, the partition function takes the form
+Z[j] =
+�
+D[φ] exp(−S[φ]/ℏ) ≃
+�
+n
+Zn[j] ,
+(6)
+where
+Zn = Fn[j] exp(−S[φn]/ℏ) ≡ exp(−Σn[j]/ℏ) ,
+(7)
+and φn are the different dominant contributions, the saddle points, which satisfy the equation of motion and minimise
+the action locally in the space of field configurations. Fn[j] are the fluctuation factors for these saddle points, that we
+will calculate at one-loop, and Σn[j] are the corresponding connected graphs generating functionals.
+The saddle points φn satisfy then
+− 1
+√g
+δS
+δφ = j ,
+(8)
+and since we consider two degenerate minima, a bubble-solution cannot form, since it would have an infinite radius
+[23, 24]. Hence we focus on homogeneous saddle points only, which can depend on the Euclidean time tough. These
+saddle points obey
+¨φ + 3˙a
+a
+˙φ − ξRφ + λ
+6 v2φ − λ
+6 φ3 = j ,
+(9)
+where a dot represents a (Euclidean) time derivative. In the adiabatic approximation, the scale factor a is assumed
+constant for the calculation of quantum fluctuations for matter, and we will therefore take ˙a = 0 = R. We discuss
+below the static saddle points and the instanton gas, with their corresponding connected graphs generating functionals
+Σ1[j], Σ2[j] and Σgas[j] respectively.
+Finally, we are interested in the tunnelling-induced effective potential, such that it is enough to consider a constant
+source j. A spacetime-dependent source is necessary for the calculation of the derivative part of the effective action
+only.
+
+4
+C.
+Static saddle points
+The static saddle points satisfy
+v2φ − φ3 = 6j
+λ ,
+(10)
+which, for j < jc ≡ λv3/(9
+√
+3), has two real solutions
+φ1(j) = 2v
+√
+3 cos
+�π
+3 − 1
+3 arccos(j/jc)
+�
+and
+φ2(j) = −φ1(−j) ,
+(11)
+with the corresponding actions
+S1[j] ≡ S[φ1(j)] =
+�
+d4x√g
+�
+¯Λ + v j −
+3
+2v2λ j2 + O(j3)
+�
+(12)
+S2[j] ≡ S[φ2(j)] = S1[���j] .
+The one-loop fluctuation factor for a static saddle point φn(j) is calculated in Appendix A, using the Schwinger
+proper time representation of the propagator. We find for the corresponding renormalised connected graphs generating
+functional
+Σn[j] =
+�
+d4x√g
+�
+¯ΛR + λR
+4! (φ2
+n − v2
+R) +
+ℏλ2
+R
+4608π2
+�
+G(φn) + 2(3φ2
+n − v2
+R)2 ln
+�
+(3φ2
+n/v2
+R − 1)/2
+��
++ jφn
+�
+(13)
+with
+G(φn) = −285v4
+R + 366v2
+Rφ2
+n − 81φ4
+n ,
+(14)
+and λR, vR, ΛR are the renormalised parameters given in Appendix A. The specific form (13), including the renor-
+malised parameters, is chosen in such a way that, in the absence of source we have
+Σn[0] =
+�
+d4x√g ¯ΛR ,
+(15)
+which makes the discussion on vacuum energy simpler. Note that, in eq.(13), the static saddle points φn can be
+expressed as in eq.(11), where the parameters can be replaced by the renormalised ones, since they satisfy the
+equation of motion [11].
+D.
+Instanton gas
+We describe here Euclidean time-dependent saddle points. In the absence of a source, they obey the following
+equation
+¨φ + ω2φ − λ
+6 φ3 = 0 ,
+(16)
+where ω = v
+�
+λ/6, which corresponds to a problem of real-time classical mechanics in the upside-down potential
+V (φ) = − λ
+24(φ2 − v2)2 ,
+(17)
+represented in Fig. 1.
+The motion starting asymptotically close to a hilltop and ending asymptotically close to the other hilltop is given
+by the known solution
+φinst(j = 0) = ±v tanh
+� ω
+√
+2(t − t1)
+�
+,
+(18)
+where t1 corresponds to the “jump”, where the instanton goes through 0, and the corresponding action is
+S[φinst(j = 0)] = a3S0
+with
+S0 ≡ 2
+√
+2
+λ ω3V0 .
+(19)
+
+5
+FIG. 1: The upside-down potential V (φ) in which the field oscillates. One instanton corresponds to the motion from
+infinitesimally close to one hilltop to infinitesimally close to the other.
+Indeed, the field spends a large (Euclidean) time close to a hilltop, with an exponentially small contribution to both
+the potential and the kinetic energy, and the main contribution to the action comes from the jump. For p jumps, an
+exact saddle point is a series of periodic oscillations between the two hills. If the motion starts exponentially close to
+a hilltop, the distance |ti+1 − ti| between two consecutive jumps is large compared to the width 2π/ω of a jump. The
+motion is then approximately described by
+φ(p)
+inst(j = 0) ≃
+± v tanh
+� ω
+√
+2(t − t1)
+�
+× tanh
+� ω
+√
+2(t − t2)
+�
+× · · · × tanh
+� ω
+√
+2(t − tp)
+�
+,
+(20)
+where the times ti are regularly spread along the Euclidean time (see Fig.2a), and the corresponding action is
+S[φ(p)
+inst(j = 0)] ≃ p a3S0 .
+(21)
+The above action remains unchanged when the jumps are shifted though, provided the condition |ti+1 − ti| ≫ 2π/ω
+is satisfied, which is called the dilute gas approximation (see Fig.2b).
+As a consequence, all the corresponding
+configurations in the path integral Z contribute as much as the exact solution of the equation of motion.
+The
+invariance of the action under the translation of jumps has a high degeneracy, making this dilute gas dominant in Z.
+We show in Appendix B that, in the presence of a source, the summation over all the p-jump saddle points leads to
+Σgas[j] ≃ 1
+2
+�
+Σ1[j] + Σ2[j]
+�
+− ℏ ln
+�
+exp( ¯N) − 1
+�
+≡ −ℏ ln Zgas[j] ,
+(22)
+where the statistical average number of jumps between the two static saddle points is
+¯N = √g00 ωT
+�
+6a3S0
+ℏπ
+e−a3S0/ℏ .
+(23)
+We note that the parameters in the latter equation can be understood as the renormalised ones, since the contribution
+of ¯N is at one-loop already.
+The exponential of ¯N appearing in the partition function is a known feature in tunnelling studies, and it arises
+from the summation over the zero modes of each (anti-)instanton (see Appendix B for details). Note that we are
+interested here in the situation where S0 is fixed and the total Euclidean time T goes to infinity, such that ¯N is
+assumed to be large. An alternative situation, relevant at finite temperature, consists in fixing T and taking S0 → ∞,
+such that ¯N → 0. This corresponds to the suppression of tunnelling, and where SSB provides a better description of
+the system [12]. The expression (23) can be understood as the total Euclidean time T multiplied by the tunnelling
+rate ω
+�
+6a3S0/ℏπ e−a3S0/ℏ.
+III.
+EFFECTIVE ACTION
+We describe here the main steps for the construction of the effective theory, as well as its energetic properties. The
+details can be found in Appendix (C).
+
+V(d)
+d
+-1.0
+1.5
+-0.5
+0.5
+1.0
+1.5
+-0.5
+-1.56
+(a) An exact saddle point configuration, corresponding to
+periodic oscillations between the two hills provided by the
+upside-down potential shown on Fig.1.
+(b) An approximate saddle point configuration with the same
+number of oscillations, but not periodic. The jumps are
+randomly distributed, but the average distance between them
+is larger than their width, such that they keep their shape and
+the action of the configuration is essentially the same as the
+action for the exact saddle point in fig.(a).
+FIG. 2: Example of exact and approximate saddle points. In the dilute gas approximation, the difference between
+the corresponding actions is exponentially small, and the partition function is dominated by the whole set of
+approximate saddle points.
+A.
+Symmetric ground state
+From the previous section, the partition function can be expressed as
+Z[j] ≃ Z1[j] + Z2[j] + Zgas[j]
+(24)
+= exp
+�
+−1
+ℏΣ1[j]
+�
++ exp
+�
+−1
+ℏΣ2[j]
+�
++ (exp( ¯N) − 1) exp
+�
+− 1
+2ℏ
+�
+Σ2[j] + Σ2[j]
+��
+,
+from which one can derive the classical field φc, which corresponds to the vacuum expectation value in the presence
+of the source j
+φc =
+−ℏ
+Z(j)√g
+δZ
+δj = −M −2 j + O(j3) .
+(25)
+
+V
+1.0
+0.5
+tw
+100 ~2
+20
+40
+60
+80
+-0.5
+-1.0V
+1.0
+0.5
+tw
+100 ~2
+20
+40
+60
+80
+-0.5
+1.07
+In the previous expression and in the limit where T → ∞, we show in Appendix C that
+M −2 =
+3
+λRv2
+R
+�
+1 + 27ℏλR
+32π2
+�
++ O(ℏ2) .
+(26)
+We note that φc is proportional to j, showing symmetry restoration: the vacuum for j = 0 is at φc = 0.
+The relation φc[j] is then inverted to
+j[φc] = −M 2φc + O(φ3
+c) ,
+(27)
+and the 1PI effective action, defined through the Legendre transform as a functional of φc, is
+Γ[φc] = −ℏ ln Z[j
+�
+φc]
+�
+−
+�
+d4x√g φc j[φc]
+(28)
+= Γ[0] + 1
+2
+�
+d4x√g M 2φ2
+c + O(φ4
+c) .
+In the previous expression, the effective action for the ground state reads
+Γ[0] =
+�
+d4x√g ¯ΛR − ℏ ln(e
+¯
+N + 1)
+(29)
+≃
+�
+d4x√g ¯ΛR − ℏ ¯N .
+To summarise the essential features of the effective action (28):
+• it is convex, since M 2 > 0, and has its ground state at φc = 0;
+• the ground state energy has a non-trivial dependence on the comoving volume, via ¯N, and is therefore not
+extensive in the usual thermodynamical sense.
+B.
+NEC violation
+For simplicity, in what follows we will drop the sub-index R and all the parameters should be understood as the
+renormalised ones.
+We focus here on the fluid provided by the ground state φc = 0. In order to obtain the energy density and the
+pressure, we need to represent Γ[0] and thus ¯N as the integral over a Lagrangian density, restoring the time dependence
+of the scale factor. This is done in Appendix C, where we show that the expression (29) can be written as
+Γ[0] =
+�
+d4x√g
+�
+¯Λ − ρ0
+e−α3
+α3/2
+�
+,
+(30)
+where
+α3 ≡ a3 S0
+ℏ
+and
+ρ0 ≡ ℏ ω
+V0
+S0
+ℏ
+�
+6
+π = λ v4
+3
+√
+3π .
+(31)
+From eq.(30), the energy density and the pressure are obtained from the components of the energy-momentum tensor
+Tµν =
+2
+√g
+δΓ[0]
+δgµν ,
+(32)
+and we find
+ρ = ¯Λ − ρ0
+e−α3
+α3/2 ,
+(33)
+p = −¯Λ + ρ0
+�
+1
+2α3/2 − α3/2
+�
+e−α3 .
+The fluid provided by the ground state therefore features the following properties:
+
+8
+• It consistently satisfies the (real-time) continuity equation ˙ρ + 3H(ρ + p) = 0;
+• It violates the NEC
+ρ + p = −ρ0 e−α3 �
+α3/2 +
+1
+2α3/2
+�
+< 0 ;
+• Assuming e−α3 ≪ 1, its equation of state has the phantom form
+w = p
+ρ ≃ −1 − ρ0
+¯Λ e−α3 �
+α3/2 +
+1
+2α3/2
+�
+< −1 .
+(34)
+We stress here an important point: the property w < −1 does not arise from a kinetic energy with the opposite sign,
+but is a consequence of tunnelling between the two degenerate bare vacua, which induces a homogeneous symmetric
+ground state.
+IV.
+FRIEDMANN EQUATIONS
+In this section we go back to Lorentzian signature. As explained in the introduction, we study the back-reaction
+of the effective theory on the metric, such that the energy-momentum tensor in the Einstein equations Gµν = κTµν
+contains the energy density and pressure given by eqs. (33), and κ is the renormalised gravity coupling. The resulting
+Friedman equations read
+H2 = κ
+3 ρ
+(35)
+¨a
+a = −κ
+6 (ρ + 3p) ,
+that we study here numerically. The first equation H2 ∝ ρ gives the initial condition ˙a0 once a0 is known, and the
+second equation provides the evolution equation for a(t). We then introduce the dimensionless time
+τ ≡ t
+�
+Λ
+3 ,
+(36)
+and we use the expressions (33) to obtain from eqs.(35)
+α′ = ±α
+�
+1 − r e−α3
+α3/2
+(37)
+α′′
+α = 1 − r e−α3
+4 α3/2 (1 − 6α3) ,
+where a prime denotes a derivative with respect to τ and
+r = κρ0
+Λ = ρ0
+¯Λ .
+(38)
+The Friedman Equations (37) are solved numerically, and we plot in Figure 3 the solutions corresponding to a fixed
+value of α(0) and different values of the parameter r. The initial condition for α′(0) is given by the negative branch
+α′(0) < 0 of the first Friedman equation, in order to describe a cosmological bounce induced by the phantom-like
+fluid. We see that such a bounce is indeed generated, after which the expansion suppresses tunnelling: the NEC is
+recovered and the metric dynamics enters a de Sitter phase, with constant H.
+V.
+CONCLUSIONS
+We have described how the energetic properties arising from tunnelling could be relevant in a cosmological context,
+starting from standard QFT and Einstein gravity. To summarise the non-perturbative mechanism described in this
+article:
+
+9
+FIG. 3: Time evolution of the scaled scale factor α (upper panel) and the scaled Hubble rate H = α′/α (lower
+panel) with initial condition α(0) = 1, for three different values of r, namely r = 2 (solid line), r = 1 (dashed line)
+and r = 0.5 (dashed-dotted line).
+(a) The effective theory taking into account tunnelling between two degenerate vacua is obtained by considering
+the contribution of different saddle points in the partition function;
+(b) As a consequence of this interplay between the two vacua ±v, the resulting true vacuum is at φc = 0, with an
+energy which is not proportional to the comoving volume;
+(c) This non-extensive feature of the vacuum energy implies NEC violation;
+(d) The NEC violation induces a cosmological bounce in the case of initial spacetime contraction, and is valid until
+the resulting expansion suppresses tunnelling, such the ANEC is satisfied.
+The adiabatic approximation is well justified in the vicinity of the cosmological bounce, but out-of-equilibrium studies
+would be necessary to include the full-time dependence of the scale factor if one wishes to look at what happens away
+from the bounce. A related improvement to this work would be to derive our results in a manifestly covariant way.
+Regarding the assumption of finite-volume FLRW space-time, this study has required a toy-model geome-
+try/topology, in the form of a 3-torus or 3-sphere, and thus still needs to be developed for phenomenological
+purposes. Also, quantum corrections in a finite volume should in principle take into account discrete momentum,
+as well as periodic boundary conditions. This is done in the framework of Casimir effect studies [36], whereas the
+present article focuses on the tunnel effect, with continuous momentum and effectively Dirichlet boundary conditions.
+A natural step further would then consider a discrete spectrum, which could be done numerically for example.
+The situation of non-degenerate minima would avoid making the finite-volume assumption, since the relevant
+instanton action (the Coleman bounce saddle point) is independent of the volume. In this case, quantum fluctuations
+for the latter saddle point would involve an imaginary part, which should be cancelled by the imaginary part induced
+by other saddle points [25], since the effective potential is real. The whole process is challenging to describe analytically
+
+7
+6
+5
+P
+4
+3
+2
+1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+t1.0
+0.5
+(1)H
+0.0
+-0.5
+-1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+t10
+in more than 0-dimensional space-time though, but is a potential avenue to explore, since it could be relevant as a
+component of Dark Energy.
+ACKNOWLEDGEMENTS
+JA would like to thank Janos Polonyi for enlightening discussions. SP thank Jose Navarro-Salas for very useful
+comments. This work is supported by the Leverhulme Trust (grant RPG-2021-299) and the Science and Technology
+Facilities Council (grant STFC-ST/T000759/1). For the purpose of Open Access, the authors have applied a CC BY
+public copyright licence to any Author Accepted Manuscript version arising from this submission.
+Appendix A: One-loop effective action in curved space-times
+In this appendix we review the main steps to obtain the one-loop effective action in curved space-times for a real
+scalar field in a double-well potential, and propagating in a curved background with Euclidean signature. We focus
+here on one saddle point only.
+For renormalisation purposes, we need to consider the bare action of this model
+S[φ, g] =
+�
+ddx√g
+�1
+2gµν∂µφ∂νφ + 1
+2ξRφ2 + λ
+4!(φ2 − v2)2 + ¯Λ + jφ
+�
+,
+(A1)
+together with the semi-classical action for gravity 1
+SG[g] = −
+�
+ddx√g
+�
+(2κ)−1R + (ϵ1R2 + ϵ2RµνRµν + ϵ3RµνρσRµνρσ)
+�
+,
+(A2)
+in d space-time dimensions, where κ = 8πG, and ¯Λ = κ−1Λ. For convenience, we have included the cosmological
+constant term in the matter sector. The inclusion of the higher curvature terms is needed for the cancellation of the
+divergences that arise in this context. In this setup, the Klein-Gordon equation for the scalar field is
+(−□E + ξR − λ
+6 v2 + λ
+3!φ2)φ + j = 0 ,
+(A3)
+where □E = gµν∇µ∇ν =
+1
+√g∂µ(√ggµν∂µ), and the scalar field can be expanded around a saddle point φ = φs + δφ.
+The associated Euclidean Green’s function for the quantum fluctuation δφ reads
+(−□E + Q)GE(x, x′) =
+1
+√g δ(4)(x − x′) ,
+(A4)
+where
+Q = λ
+2 φ2
+s − λv2
+6
++ ξR .
+(A5)
+The one-loop correction to the classical action can be written in terms of the Green’s function as [37]
+Σ[φs, g] = SG[g] + S[φs, g] − 1
+2ℏ ln Det GE
+(A6)
+≡ SG[g] + S[φs, g] + Σ(1)[φs, g] .
+For general background configurations, the Green’s function is unknown. However, an approximated expression for
+the quantum contribution Σ(1)[φs, g] in the case of slowly varying background fields φs and g can be computed using
+the proper-time formalism as follows (see Refs. [38, 39] for a detailed explanation).
+The DeWitt-Schwinger representation of the propagator GE(x, x′) is given by
+GE(x, x′) =
+� ∞
+0
+ds H(x, x′; s) ,
+(A7)
+1 We note that the Euclidean form of the Lagrangian differs with a minus sign with respect to its Lorentzian form.
+
+11
+where the kernel H(x, x′; s) obeys a diffusion equation with appropriate boundary conditions [40]. For the one-loop
+connected graph, it translates into
+Σ(1)[φs, g] = ℏ
+2
+�
+ddx√g
+� ∞
+0
+ds
+s H(x, x; s) .
+(A8)
+The kernel H(x, x′; s) admits, in general, an asymptotic expansion in terms of the Schwinger proper-time parameter
+[41]. At coincidence x′ → x this expansion reads
+H(x, x; s) =
+e−m2s
+(4πs)d/2
+∞
+�
+k=0
+ak(x) sk .
+(A9)
+where ak(x) are the so-called the deWitt coefficients and d is the space-time dimension. The first few coefficients are
+[40, 42]
+a0 = 1 ;
+(A10)
+a1 = 1
+6R − Q ;
+(A11)
+a2 = − 1
+180RαβγδRαβγδ −
+1
+180RαβRαβ − 1
+30□ER + 1
+6□EQ + 1
+2Q2 − 1
+6RQ + 1
+72R2 .
+(A12)
+This expansion captures, in its leading orders, the UV divergences (s → 0) of the theory and it is routinely used for
+renormalisation in the context of QFT in curved spaces.
+The expansion above (A9) has an important property: it admits an exact resummation [43, 44]
+H(x, x; s) = e−M2s
+(4πs)d/2
+∞
+�
+k=0
+bk(x) sk ,
+(A13)
+with
+M2 = Q − 1
+6R ,
+(A14)
+such that, the new coefficients bk(x) do not contain any term that vanish when Q and R are replaced by zero. For
+example, for the first resummed deWitt coefficients we have
+b0 = 1 ;
+(A15)
+b1 = 0 ;
+(A16)
+b2 = − 1
+180RαβγδRαβγδ −
+1
+180RαβRαβ − 1
+30□ER + 1
+6□EQ .
+(A17)
+Therefore, the resummed expansion becomes a derivative expansion in the field φs and the metric, physically mean-
+ingful in the case of slowly varying background fields. Then, it is possible to truncate the expansion at a given order
+N - the order of derivatives - to obtain an approximated expression for the one-loop connected graph
+Σ(1)[φs, g] = ℏ
+2
+�
+ddx√g
+� ∞
+0
+ds
+s
+e−M2s
+(4πs)d/2
+N
+�
+k=0
+bk(x) sk .
+(A18)
+The expression above is divergent for d = 4 and can be renormalised using dimensional regularization. For arbitrary
+dimension d, the proper-time integrals can be performed to give
+Σ(1)[φs, g] =
+ℏ
+(4π)d/2
+�M
+µd
+�d−4 �
+ddx√g
+N
+�
+k=0
+bk(x)Md−2k Γ
+�
+k − d
+2
+�
+.
+(A19)
+We have introduced a renormalisation mass parameter to proceed with dimensional regularization in what follows.
+Truncating the sum at N = 2 and expanding around d → 4 we find
+Σ(1) = ℏ
+�
+d4x√g
+� M4
+64π2
+�
+ln
+�M2
+µ2
+�
+− 3
+2
+�
++
+b2
+32π2 ln
+�M2
+µ2
+��
+,
+(A20)
+
+12
+where M2 > 0 since we quantise about stable saddle points and the curvature effects are expected to be small. In
+the above expression, the divergences have been absorbed in the scale parameter µ, which is defined by
+ln µ2 = ln
+�
+4πµ2
+d
+�
+− γ −
+2
+d − 4
+(finite when d → 4) .
+(A21)
+From these expressions, we can directly obtain the renormalised values of the coupling constants of the problem
+(see, for example Ref. [45]).
+In our particular problem, we are assuming an adiabatic expansion of the universe, and that quantum processes
+under consideration occur at equilibrium. Hence, we neglect the curvature of space-time. Hence we are only interested
+in the couplings λ, v, ¯Λ. For simplicity, we will follow [46], and apply the renormalisation conditions at the same
+scale for all bare parameters, namely,
+3∂2L
+∂φ2
+���
+φ=±vR,g=η = λR v2
+R ,
+∂2L
+∂φ4
+���
+φ=±vR,g=η = λR ,
++L
+���
+φ=±vR,g=η = ¯ΛR ,
+(A22)
+where η is the Euclidean flat metric and Σ =
+�
+d4x√g L. From these conditions we obtain
+δλ = 3λ2
+R
+32π2
+�
+3 + ln(v2
+RλR
+3µ2 )
+�
+,
+(A23)
+δv2 = v2
+RλR
+16π2
+�
+10 − ln(v2
+RλR
+3µ2 )
+�
+,
+(A24)
+δ¯Λ = v4
+Rλ2
+R
+1152π2
+�
+−3 + 2 ln(v2
+RλR
+3µ2 )
+�
+,
+(A25)
+where we define λR = λ + ℏ δλ, v2
+R = v2 + ℏ δv2, and ¯ΛR = ¯Λ + ℏ δ¯Λ. Inserting these results in (A6) and assuming
+R = 0 and φs static, we obtain the final renormalised connected graph given in Sec. II C.
+For completeness, we also give the renormalised values of κ−1 and ξ. The renormalisation conditons we impose are
+− 2 ∂L
+∂R − ξRφ2���
+φ=±vR,g=η = κ−1
+R ,
+∂3L
+∂R∂φ2
+���
+φ=±vR,g=η = ξR ,
+(A26)
+that lead to
+δξ = λR(6ξR − 1)
+192π2
+�
+3 + ln(v2
+RλR
+3µ2 )
+�
+,
+(A27)
+δ(κ−1) = v2
+RλR(6ξR − 1)
+2304π2
+�
+11 + ln(v2
+RλR
+3µ2 )
+�
+.
+(A28)
+Appendix B: Quantisation over instanton configurations
+In Section II D we describe few features of the gas of instantons for a vanishing source. In the presence of an
+infinitesimal source j ≪ jc, the jump is not modified, and what changes is the position of the asymptotically ”flat”
+parts of the instantons, which now go from one saddle point φi(j) to the other, instead of going from one vacumm
+±v to the other ∓v. We have then, instead of eq.(19),
+S[φinst(j)] ≃ a3S0 + 1
+2
+�
+S1[j] + S2[j]
+�
+,
+(B1)
+since on average the configuration spends half the Euclidean time exponentially close to φ1(j) and the other half close
+to φ2(j). The contribution of one instanton Finst exp(−S[φinst]/ℏ) to the partition function is the product of the
+following contributions
+• The ”flat” part close to each static saddle point, leading to the fluctuation factor Fi about each of the static
+saddle points, for half of the total Euclidean time
+�
+F1F2e−(S1+S2)/(2ℏ) = exp
+�
+− 1
+2ℏ
+�
+Σ1[j] + Σ2[j]
+��
+,
+(B2)
+where Σn[j] is given in eq.(13).
+
+13
+• Fluctuations above one jump which, discounting the zero mode corresponding to the translational invariance of
+the jump, lead to the factor (see [27, 47])
+�
+6a3S0
+ℏπ
+;
+(B3)
+• The zero mode corresponding to the position of the jump, which can happen at any Euclidean time between 0
+and T, and thus gives the extra factor
+ω
+� T
+0
+√g00 dt = √g00 ωT .
+(B4)
+Note that the summation over the different positions of the jump is done with the comoving proper time, since
+the jump is observed by the comoving observer. Here, S0 and ω are defined with the renormalised parameters.
+All together, the contribution of one instanton to the partition function is
+Finst exp
+�
+−S[φinst]
+ℏ
+�
+= √g00 ωT
+�
+6a3S0
+ℏπ
+exp
+�
+−a3 S0
+ℏ − 1
+2ℏ
+�
+Σ1[j] + Σ2[j]
+��
+.
+(B5)
+For a p-jump saddle point in the dilute gas approximation, and where the width of an instanton is negligible compared
+to the total Euclidean time T, the classical action is
+S[φp
+inst(j)] ≃ pa3S0 + 1
+2
+�
+S1[j] + S2[j]
+�
+.
+(B6)
+Also, whereas the first jump can happen at any time t1 ∈ [0, T], the jump i can happen at a time ti ∈ [ti−1, T] only,
+such that the degeneracy of a p-jump configuration leads to the factor [27]
+p
+�
+i=1
+�
+ω
+� T
+ti−1
+√g00 dti
+�
+= 1
+p!(√g00 ωT)p
+(with t0 = 0) .
+(B7)
+Summing over all the possibilities for p, we obtain the final expression for the dilute gas contribution to the partition
+function
+exp
+�
+−1
+ℏΣgas[j]
+�
+=
+∞
+�
+p=1
+1
+p!(√g00 ωT)p
+�6a3S0
+ℏπ
+�p/2
+exp
+�
+−pa3 S0
+ℏ − 1
+2ℏ
+�
+Σ1[j] + Σ2[j]
+��
+(B8)
+= exp
+�
+− 1
+2ℏ
+�
+Σ1[j] + Σ2[j]
+�� �
+exp
+�
+√g00 ωT
+�
+6a3S0
+ℏπ
+e−a3S0/ℏ
+�
+− 1
+�
+.
+Appendix C: Effective action, energy density and pressure
+We give here details on the derivation of the one-loop effective action. We start from the partition function
+Z[j] = Z1[j] + Z2[j] + Zgas[j]
+(C1)
+= e−Σ1/ℏ + e−Σ2/ℏ + (e
+¯
+N − 1)e−(Σ1+Σ2)/2ℏ ,
+where Σ2[j] = Σ1[−j] which, for small source, can be expanded as
+Σ1,2[j] =
+�
+d4x√g
+�
+¯ΛR ± σ(1) j + 1
+2σ(2) j2 + O(j3)
+�
+,
+(C2)
+with
+σ(1) = vR − ℏ 9λRvR
+32π2 ,
+σ(2) = −
+3
+v2
+RλR
+− ℏ
+81
+32π2v2
+R
+.
+(C3)
+
+14
+The classical field φc is
+φc =
+−ℏ
+Z(j)√g
+δZ
+δj = −M −2 j + O(j3) ,
+(C4)
+with
+M −2 = −σ(2) + V4
+ℏ
+2
+(e ¯
+N + 1)σ2
+(1) =
+3
+λRv2
+R
+�
+1 + 2A
+3 + ℏλR
+27
+2π2
+� 1
+16 − A
+��
++ O(ℏ2) ,
+(C5)
+and
+V4 =
+�
+d4x√g
+,
+A =
+V4 ω4
+R
+ℏλR(e ¯
+N + 1) .
+(C6)
+The relation φc[j] is then inverted to j[φc], in order to define the 1PI effective action as the Legendre transform
+Γ[φc] = −ℏ ln Z[j
+�
+φc]
+�
+−
+�
+d4x√g φc j[φc] .
+(C7)
+An expansion in the classical field finally gives
+Γ[φc] = Γ[0] +
+�
+d4x√g M 2
+2 φ2
+c + O(φ4
+c) ,
+(C8)
+with
+M 2 =
+�
+−σ(2) + V4
+ℏ
+2
+e ¯
+N − 1σ2
+(1)
+�−1
+(C9)
+= λRv2
+R
+3
+�
+1
+1 + 24A − ℏλR
+27
+32π2
+1 − 16A
+(1 + 24A)2
+�
++ O(ℏ2) ,
+and
+Γ[0] =
+�
+d4x√g ¯ΛR − ℏ ln(e
+¯
+N + 1) ≃
+�
+d4x√g ¯ΛR − ℏ ¯N .
+(C10)
+In the limit T → ∞ we obtain then
+M 2 = λRv2
+R
+3
+�
+1 − ℏλR
+27
+32π2
+�
++ O(ℏ2) .
+(C11)
+The next step is the analysis of the energy density and pressure for the ground state. The stress-energy tensor can
+be obtained from the definition
+T E
+µν =
+2
+√g
+δΓ(0)
+δgµν .
+(C12)
+where we have explicitly written the super-index E as a reminder that we are working in Euclidean signature. Because
+of homogeneity and isotropy, the stress-energy tensor can be decomposed as
+T E
+µν = diag(−ρ, a2p, a2p, a2p) ,
+(C13)
+so that we directly obtain
+ρ = −T E
+00 = − 2
+√g
+δΓ(0)
+δg00
+���
+g00=1 ,
+p = g11T E
+11 =
+2
+a2√g
+δΓ(0)
+δg11
+���
+g11=a−2 .
+(C14)
+In order to express ¯N as a Lagrangian density we restore the time dependence of the scale factor with the replacement
+√g00 T f(a) →
+� T
+0
+dt √g00 f(a)
+(C15)
+
+15
+and we express the cell 3-volume at t = t0 as
+V0 =
+�
+d3x a3(t0) =
+�
+d3x
+if we choose
+a(t0) = 1 .
+(C16)
+The effective action for the ground state for ωRT ≫ 1 [see Eq. (C10)] can then be expressed as
+Γ[0] ≃
+�
+d4x√g ¯ΛR − ℏωR
+�
+6S0
+ℏπ
+� T
+0
+dt√g00
+� d3x
+V0
+a3/2 e−a3S0/ℏ
+(C17)
+=
+�
+d4x√g
+�
+¯ΛR − ρ0
+e−a3S0/ℏ
+�
+a3S0/ℏ
+�
+,
+where
+ρ0 ≡ ωRS0
+V0
+�
+6
+π = λRv4
+R
+3
+√
+3π ,
+(C18)
+and where S0 is defined with the renormalised parameters.
+From Eqs. (C14) and (C17) we can easily obtain the energy density and the pressure, namely
+ρ = −T E
+00 = −
+2
+√g
+δΓ(0)
+δg00
+����
+g00=1
+= +¯ΛR − ρ0
+e−a3S0/ℏ
+�
+a3S0/ℏ
+,
+(C19)
+p = g11T E
+11 =
+2
+a2√g
+δΓ(0)
+δg11
+����
+g11=a2
+= −¯ΛR + ρ0
+�
+1
+2
+�
+a3S0/ℏ
+−
+�
+a3S0/ℏ
+�
+e−a3S0/ℏ .
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+

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+arXiv:2301.05326v1  [astro-ph.GA]  12 Jan 2023
+Kinematics and Origin of Gas
+in the Disk Galaxy NGC 2655
+O.K. Silchenko1, A.V. Moiseev2,1, A.S. Gusev1, and D.V. Kozlova3
+Sternberg Astronomical Institute of the Lomonosov Moscow State University, Moscow, Russia1
+Special Astrophysical Observatory of the Russian Academy of Sciences, Nizhnij Arkhyz, Russia2
+Leibniz-Institut fur Astrophysik (AIP), Ander Sternwarte 16, 14482 Potsdam, Germany3
+The new observational data concerning distribution, excitation, and kinematics of the ionized
+gas in the giant early-type disk galaxy NGC 2655 obtained at the 6m telescope of the Special
+Astrophysical Observatory (SAO RAS) and at the 2.5m telescope of the Caucasian Mountain
+Observatory (CMO SAI MSU) are presented in this work. The joint analysis of these and
+earlier spectral observations has allowed us to make a conclusion about multiple nature of the
+gas in NGC 2655. Together with a proper large gaseous disk experiencing regular circular
+rotation in the equatorial plane of the stellar potential of the galaxy for billions years, we
+observe also remnants of a merged small satellite having striked the central part of NGC 2655
+almost vertically for some 10 million years ago.
+Keywords: galaxies: early-type—galaxies: evolution—galaxies: starformation—galaxies:
+individual: NGC 2655
+1
+
+1
+INTRODUCTION
+The morphological type of lenticular galaxies was introduced by Hubble (1936) as a transitional
+type between ellipticals and spirals. However, having a large-scale stellar disk in their structure,
+lenticulars did not show noticeable star formation in it, opposite to spirals. It is very early that
+a hypothesis was proposed that star formation does not occur in the disks of lenticular galaxies,
+because there is no gas there; and S0s have no gas, because it was somehow ”removed”, for
+example, by interaction with hot intergalactic medium in a cluster (Gunn and Gott, 1972;
+Larson et al., 1980). However, since then the paradigm of the spiral (disk) galaxy evolution
+has changed since then, it became clear that the entire evolution of disk galaxies is governed by
+an inflow of cold gas from outside compensating for any losses of it in the disk, in particular,
+the losses due to star formation (Tacconi et al., 2020). Also, deep radio observations, both
+large-scale surveys like ALFALFA (Grossi et al., 2009) and targeted studies of specific samples
+of early-type galaxies, such as the ATLAS-3D survey (Serra et al., 2012), showed that almost
+half of the field lenticular galaxies has massive extended gaseous disks. Why does not the same
+star formation take place in these disks as that in the disks of spiral galaxies?
+Observations of gas kinematics in field lenticular galaxies have always shown impressive
+fraction of decoupled rotation of gas and stars – from 24% (Kuijken et al., 1996) in earlier
+long-slit studies up to 36% (Davis et al., 2011) and even up to half of all lenticular galaxies
+in the extremely sparse environment (Katkov et al., 2015).
+We have previously concluded
+(Silchenko et al. 2019) that the suppressed star formation in gas-rich lenticular galaxies may
+be due to the off-plane inflow of the accretion flow: the gas falling into the potential well of
+the stellar disk suffers shocks, is heated, and becomes unable to form stars. We tested this
+hypothesis with a sample of 18 lenticular galaxies having extended gaseous disks, by observing
+them through panoramic spectroscopy, with the Fabry-Perot scanning interferometer of the
+6m SAO RAS telescope. We have constructed 2D line-of-sight velocity distributions and have
+2
+
+traced the orientation of the gas rotation plane in space along the galactic radius. Indeed, star
+formation (particularly the star formation rings in lenticular galaxies) appeared to locate only
+at the radii, where the gas lies onto the plane of the stellar disk. On the contrary, in inclined
+gaseous disks,star formation does not proceed (Silchenko et al., 2019). One of the targets in
+our sample in this work was a nearby giant lenticular galaxy NGC 2655. Figure 1 presents its
+images provided by the ground-based photometric observations taken from the BASS survey
+and by high spatial resolution composite observations from the Hubble Space Telescope.
+Figure 1: The images of NGC 2655 in composite colors: the left plot – the deep broad-band
+image of the galaxy taken from the DESI Legacy Imaging Surveys resource (Dey et al. 2019),
+the right plot – the image of the central part of the galaxy obtained in broad-band filters by
+the Hubble Space Telescope. At the the right plot one can see asymmetric dust rings produces
+by the projection of the circumnuclear polar disk.
+NGC 2655 is a giant disk galaxy in the center of a group: at a currently accepted distance
+3
+
+to the galaxy of 24.4 Mpc (the scale is 118 pc/arcsec), its absolute magnitude is MK = −25
+(LEDA and NED), and the mass of the stellar population is 2 · 1011 solar masses (Bouquin
+et al., 2018). The group includes seven galaxies brighter than MB = −15, all of them are of
+the late type (Garcia 1993). With this configuration, one would expect that the whole gas
+content of NGC 2655 could result from accumulating the surrounding dwarfs by the central
+galaxy. Indeed, NGC 2655 is abundant in neutral hydrogen: according to the earliest surveys,
+up to 3–6 billion solar masses of neutral hydrogen have been found in the galaxy (Lewis and
+Davies, 1973). It forms a giant disk with a diameter of five times the diameter of the stellar disk
+(Huchtmeier and Richter, 1982). The integrated star formation rate (SFR) estimated from the
+ultraviolet fluxes of the galaxy according to the data produced by the GALEX space telescope
+is 0.08 solar masses per year (Bouquin et al., 2018), which places the galaxy significantly below
+the Main Sequence classifying it as a ”galaxy with quenched star formation” (Cortese et al.
+2020). At the same time, it should be noted that such SFR is anomalously low for the observed
+abundance of H I (Catinella et al. 2018). Detailed investigation of the spatial distribution of the
+neutral hydrogen density (Shane and Krumm 1983, Sparke et al. 2008) reveals the extension
+of the gaseous disk in a position angle of ∼ 110◦; we are going to compare this orientation
+with the parameters of the orientation of the stellar disk in Discussion in the present paper.
+As for the kinematics of the stars and gas, mapped for the central part of the galaxy through
+the panoramic spectroscopy, the gas demonstrates a polar rotation in the center with respect
+to the stars (Silchenko and Afanasiev, 2004; Dumas et al., 2007).
+Another feature which is worth to be taken into account is the active nucleus of NGC
+2655. Most researchers consider the nucleus of NGC 2655 as a Seyfert type II following our
+conclusion (Silchenko and Burenkov, 1990); but, for example, Keel and Hummel (1988) noted
+a strong emission line [OI]λ6300 in the nucleus spectrum and classified it as a LINER. The
+NGC 2655 nucleus reveals a noticeable flux in X-ray including hard X-ray range (Terashima et
+al., 2002). High-resolution mapping of the NGC 2655 nucleus in the radio continuum detects
+4
+
+a source with a steep spectrum which is compact both at wavelengths of 6 cm and 20 cm
+(Hummel et al., 1984); and from the nucleus a jet comes out in the west-east direction, which
+curves farther from the nucleus to the north-south direction (Ho and Ulvestad, 2001). Perhaps
+it is the jet that excites another compact radio source, at 15′′ (1.7 kpc) to the south-east from
+the nucleus, which demonstrates the same steep spectrum as the nucleus (Keel and Hummel,
+1988).
+NGC 2655 is a testbed case of highly inclined rotation of gas in the absence of any star
+formation in a gas-rich S0, which is of particular interest for us. However, the large-scale pattern
+of the velocity distribution in the extended gaseous disk of NGC 2655 cannot be understood
+within a simple geometric model of a flat inclined rotation plane. Both the velocities and the
+brightness distribution of the emission lines in this galaxy reveal a very complex pattern. We
+have undertaken some additional observations and are now ready to look into the details of
+how and when the gas has come to NGC 2655.
+2
+NEW OBSERVATIONS
+We have already devoted several papers to the galaxy NGC 2655 (Silchenko and Burenkov,
+1990; Silchenko and Afanasiev, 2004; Silchenko et al., 2019), and we have since a tremendous
+collection of the spectroscopic data obtained earlier with the 6-m BTA telescope. However,
+some incomprehensible moments remained in the interpretation of the ionized gas kinematics
+and, in order to clarify the whole picture, we decided to obtain additionally data.
+2.1
+Mapping in Emission Lines
+We obtained an image of the galaxy with the NBI camera (Shatsky et al., 2020) in a narrow
+Halp filter centered onto the complex of bright ionized-gas emission lines Hα+[NII]λλ6548,6583,
+having the transition peak at 656 nm, with the 2.5-m telescope of the Caucasus Mountain
+Observatory of SAI MSU (Shatsky et al., 2020) on January 10, 2018. The seeing during the
+5
+
+observations was 2.5′′. The center wavelength of the filter used was 6560 ˚A, the bandwidth was
+77˚A, so both the [NII]λλ6548,6583 doublet lines and the hydrogen Balmer line Hα fell there.
+At the same time, a feature of NGC 2655 is that the [NII]λ6583 line is stronger than the Hα line
+everywhere through the body of the galaxy (Silchenko and Burenkov, 1990). The total exposure
+of the galaxy image obtained in the emission lines was 25 minutes. The image scale was 0.155′′
+per pixel. In addition to photometry in the narrow Halp filter, the galaxy was also exposed in
+the neighboring continuum for 20 minutes (through the filter with a width of 100˚A centered
+on 6430˚A), so that after subtracting the image in the continuum from the image obtained in
+the Halp filter, it would be possible to obtain a proper intensity distribution of the emission
+lines. Figure 2 shows the result of this procedure together with the color map calculated from
+the broadband photometry in the BASS survey (taken from the Legacy Survey resource, Dey
+et al. 2019). The morphology of the image in the emission lines represents a narrow loop,
+the center of which does not coincide with the center of the galaxy, plus a compact emission-
+line region to south-east from the nucleus, which was previously detected in radiocontinuum
+emission (indicated in our picture as ESE). A red (dusty) loop outlines the inner edge of the gas
+emission loop and is especially bright to the south of the center of NGC 2655. It is apparently
+associated with shock fronts generated by the collision of the polar nuclear gaseous disk with
+proper gas of the galaxy, probably lying in the main plane of the galactic disk.
+2.2
+Long-slit Spectroscopy
+Additional long-slit spectroscopic data were obtained on May 26, 2022, at the BTA, the 6m
+SAO RAS telescope, with the SCORPIO-2 multi-mode focal reducer (Afanasiev and Moiseev,
+2011). The VPHG1200@540 grism was used with a sensitivity maximum at 5400 ˚A providing
+the full optical range of spectroscopic observations in the wavelength range of 3650–7300 ˚A
+with a resolution of about 5 ˚A. The slit was posed in two position angles: to include the ”radio
+loud” ESE compact emission region (Fig. 2) and to catch the top of the northern part of the
+6
+
+Figure 2: The central part of NGC 2655: the left plot – the g − r color image derived from the
+data of the BASS survey, the right plot – the image through the narrow-band filter Halp, which
+includes the emission lines Hα+[NII]λλ6548,6583, according to our data obtained at the 2.5m
+telescope of CMO SAI MSU, after continuum subtracting. Some particular regions seen in the
+emissione lines are marked for further spectral analysis and discussion.
+circumnuclear emission loop; the exposure times were 1600 sec and 800 sec respectively. The
+seeing quality during the spectroscopic observations in 2022 was 2.4′′. These long-slit cross-
+sections, together with the cross-sections at the position angles of PA = 102◦ and PA = 0◦,
+previously obtained with the same instrument and the same grism, were used to measure the
+fluxes of various emission lines and their ratios for the selected regions at different distances
+from the center of the galaxy, and also to trace the line-of-sight velocities of the gas and the
+stars.
+3
+EXCITATION OF THE IONIZED GAS
+Previously it was noted more than once (e.g., Sil’chenko and Burenkov 1990, Keel and Hummel
+1988) that the strong emission lines in the spectrum of the NGC 2655 nucleus show flux ratios
+characteristic of a Seyfert type II active nucleus or a LINER one. Moreover, Keel and Hummel
+7
+
+(1988), analyzing their spectrum of the ESE clump, modest as concerning the spectral range
+and the S/N ratio, have suspected that the spectrum of the ESE clump which is located in 1.8
+kpc from the nucleus, is very similar to the nuclear spectrum in terms of the pattern of line flux
+ratios. Since the limitations on the energy of the nucleus did not allow explaining the ionized
+gas of the ESE clump as excited by the radiation of the central engine of the active nucleus,
+it was proposed that the gas excitation source here is a shock wave from the active nucleus
+jet which, according to radio interferometry, seems to be directed at the appropriate position
+angle.
+We obtained rather deep spectra with the 6-m BTA telescope at four different slit ori-
+entations. Measurements of the flux ratios of the emission lines in these four spectra showed
+that the characteristic flux ratio dominated by the highly-excited [OIII]λ5007 line is observed
+throughout the disk of NGC 2655, not only in the ESE clump but also in the polar central loop
+(clumps N, W1, and S), and in the regular gaseous disk of NGC 2655 up to the distance of
+8 kpc from the center. Figure 3 shows the diagnostic diagrams – the so called BPT-diagrams
+proposed by Baldwin et al. (1981) for diagnozing a gas ionization source – to compare the ratio
+of the high-excitation [OIII]λ5007 line to the nearest Balmer hydrogen line Hβ, and the ratio
+of the low-excitation [NII]λ6583 line to the neighboring Balmer hydrogen line Hα, for some
+selected areas of NGC 2655. The red dotted and green dashed lines separate the area of the
+emission regions excited by young stars (to the left and below the line) from other excitation
+mechanisms, according to the papers by Kauffmann et al. (2003) and Kewley et al. (2001)
+respectively. Other excitation mechanisms are the ionization either by the power-law spectrum
+of the active nucleus or by a shock wave: the BPT-diagram does not makes it possible to dis-
+tinguish between these two mechanisms. Since the regions under study are located at different
+distances from the active nucleus, from 1 to 8 kpc, and the line ratios are similar for all them,
+we think that we are dealing with gas excitation by a shock wave. Seven of the eight regions
+studied contain the gas likely excited by shock wave. Although the areas of excitation by shock
+8
+
+Figure 3: The diagnostic BPT-diagrams to determine a ionized-gas excitation source presented
+for four long-slit cross-sections of NGC 2655. The slit PAs are indicated in the upper left corner
+of each plot, the distance from the center (and the direction along the slit, north or south, east
+or west) are given for every point (emission-line region). The dotted red line and the green
+dashed line separate the areas for emission regions excited by young stars (to the left and
+down) and other excitation mechanisms according to Kauffmann et al. (2003) and Kewley et
+al. (2001), respectively. The dashed-dot fat lines show the models of shock excitation for the
+gas with solar metallicity and the typical electronic density of n = 1 cm−3 according to Allen
+et al. (2008). Along every model broken line the shock velocity rises from bottom to top, from
+200 km/s to 1000 km/s; the right broken line corresponds to the shock wave propagating in
+low-density environment, and the left one – to the shock model with precursor.
+9
+
+waves and by a Seyfert nucleus overlap in the BPT-diagrams, in this case we are talking about
+the gas excitation at a large distance from the center, and already Keel and Hummel (1988)
+have estimated that the radiation from the active nucleus of NGC 2655 is not enough even to
+excite the ESE region at 15′′ from the center, not to mention more distant regions. At the
+orientation PA = 102◦ one can see how the shock wave slows down with distance from the
+center: if we compare the line ratios neasured by us with the Allen et al. (2008) models, then
+from the point r = 20′′ to the point r = 60′′, the velocity of the shock wave falls down by
+some 150 km/s. Only a single region, at 7 kpc south of the center in PA = 158◦, is excited
+by young stars. This compact region is located at the periphery of the outer disk and is also
+visible in the ultraviolet (Fig. 4). Since the gas in this region is ionized by young stars, we can
+estimate its metallicity from the strong-line flux ratios calibrated using the HII region spectra
+modeled in detail. We used two popular sources of such calibrations and obtained estimates for
+the oxygen abundance in the outer gas for NGC 2655: 12 + log (O/H) = 8.58 ± 0.18 dex by the
+indicator N2 and 12+log (O/H) = 8.58±0.16 dex by the indicator O3N2 (Marino et al. 2013),
+or 12 + log (O/H) = 8.71 ± 0.18 dex by the indicator N2 and 12 + log (O/H) = 8.79 ± 0.21 dex
+by the indicator O3N2 (Pettini and Pagel, 2004). Despite the low accuracy of these estimates,
+we can still confirm that the metallicity of the gas is approximately solar, and this is at the
+periphery of the galaxy disk!
+4
+THE DETAILED GAS KINEMATICS
+Earlier, we noted more than once the polar rotation of the ionized gas in the central region
+of NGC 2655 (Silchenko and Afanasiev, 2004; Silchenko et al., 2019).
+However the actual
+pattern of gas kinematics throughout the entire galaxy can be much more complicated than
+simply warped rotation plane. The neutral hydrogen outside the stellar disk rotates regularly,
+in a circular manner according to the apparent orientation of the HI disk, with a kinematical
+major axis close to PA = 110◦; Sparke et al. (2008) proposed a model with a smooth turn of
+10
+
+Figure 4: The ultraviolet maps of NGC 2655 according to the GALEX data: the left – the FUV
+map, λ ≈ 1500 ˚A, it the right – the NUV map, λ ≈ 2300 ˚A.
+the gaseous disk when going toward the center of the galaxy. Our data on the ionized gas in
+the outermost regions of the disk, at R > 40′′, also seem to agree with the stellar kinematics
+(Silchenko et al., 2019). However, a lot of details in the distribution of the emission-line surface
+brightness in Fig. 2 rather indicates not a smooth warp of the gaseous disk but the presence
+of several gas subsystems with different kinematics at the line of sight. This last hypothesis is
+also consistent with the shock excitation of the gas throughout the disk of NGC 2655.
+Using the benefit of rather high spectral resolution of our data obtained with the Fabry-
+Perot scanning interferometer, as a second step of our analysis of these data, we decided to take
+a closer look at the line profiles; the line analyzed is the [OIII]λ5007 emission line scanned in the
+narrow spectral range over the entire body of NGC 2655 with the Fabry-Perot interferometer
+(FPI). The line profiles appeared to be complex and multi-component. Figure 5 presents the
+examples of the Gaussian line fitting for the loop areas marked as N, W1, W2, and S in Fig. 2.
+Let us note that although the FPI instrumental profile differs from the pure Gaussian one and
+can be rather described by a Voigt profile, but in the case of the given FPI, the observed line
+profiles differ little from the Gaussian one which can be clearly seen in Fig. 5.2. In every region
+11
+
+Figure 5: The Gauss-analysis of the [OIII]λ5007 emission-line profiles for the compact emission-
+line regions designated in Fig. 2 as N (the upper left), W1 (the upper right), W2 (the bottom
+left), and S (the bottom right). The three former regions reveal the presence of two velocity
+components at the line of sight each: 1196±12 km/s and 1371±16 km/s (N), 1273±33 km/s and
+1401±37 km/s (W1), 1560±6 km/s and 1431±6 km/s (W2), respectively. The southern loop
+clump reveals three velocity components, 1539±11 km/s, 1667±32 km/s, and 1371±16 km/s.
+12
+
+Figure 6: The Gauss-analysis of the emission-line profiles for the clump ESE: the [OIII]λ5007
+line has the velocity components 1499±99 km/s and 1737±377 km/s, according to the Fabry-
+Perot data analysis; and the long-slit data gives 1517 ± 35 km/s and 1730 ± 295 km/s for the
+Hα profile, and 1490 ± 36 km/s and 1677 ± 138 km/s for the nitrogen doublet profile.
+N, W1, W2, and S, we can distinguish at least two components with different line-of-sight
+velocities. In the N and S regions, the stronger components imply the polar rotation of the
+loop; but there are also weaker components demonstrating line-of-sight velocities close to the
+systemic velocity of the galaxy, 1400 km/s, that is expected for the gas at the minor axis of the
+disk. Obviously, the weak components belong to the gaseous disk rotating in the plane of the
+stellar disk, whose isophote major axis (and the line of nodes) is aligned close to the west-east
+direction.
+For the radio-loud ESE emission region located at 1.8 kpc southeast of the nucleus, Fig. 6
+shows the results of the Gaussian fitting for three lines: for the oxygen line derived from the
+Fabry-Perot data and for the hydrogen Hα and the nitrogen doublet according to the long-slit
+spectroscopy data. Although the weak component is measured here with low accuracy, but for
+all three elements it occurs that in the ESE region there is the gas with a line-of-sight velocity
+of about 1700 km/s; it is larger by 300 km/s than the systemic velocity of the galaxy. The gas
+with the similar velocity is observed at the southwestern edge of the gaseous disk according
+to the Fabry-Perot velocity field (Silchenko et al., 2019), and this velocity does not match any
+13
+
+circular rotation models. Apparently, as regarding the ESE clump, this may be a compact
+remnant of a satellite which has hit the NGC 2655 disk with a high impact velocity of 400–
+500 km/s almost at a right angle to the stellar disk. The whole configuration with a destroyed
+companion and a polar circumnuclear loop looks like the destroyed Milky Way companion dSgr
+stretched into a polar stream in our Galaxy (Ibata et al., 2001; Laporte et al., 2018). And the
+NGC 2655 proper gas, which was hit in the center by the fallen companion, should have lost
+momentum in the shock wave and inflow into the nucleus; perhaps, this is what fuelled the
+current activity of the nucleus.
+5
+DISCUSSION
+5.1
+Structure and Stellar Kinematics of NGC 2655
+NGC 2655 is a giant early-type disk galaxy. It is commonly accepted that such galaxies should
+have a very large dominant bulge. Indeed, a detailed morphological analysis and decomposition
+of the galaxy image into components undertaken as a part of the S4G survey of galaxies (Sheth
+et al., 2010) showed that the disk contributes no more than 42% to the near-infrared luminosity –
+and then to the stellar mass (Salo et al., 2015). According to this decomposition, the exponential
+disk starts to dominate in the surface brightness at the radii of R > 50′′, while closer to the
+center, the surface brightness profile represents a combined contribution of the bulge and bar.
+Why have the Salo et al. (2015) team decided that NGC 2655 has a bar, even though the galaxy
+is not classified as SB in any catalog? This is due to the fact that the major-axis orientation
+of the isophotes of the inner components – the one with the PA1 = 82◦, and the other with
+PA2 = 85.6◦, – differ from the orientation of the outermost disk isophotes, PA0 = 110◦, which
+is commonly treated as the orientation of the line of nodes (under the assumption of the round
+intrinsic shape of the disk).
+As a result, the NGC 2655 image deprojection undertaken in
+the S4G survey by the Salo et al.
+(2015) team exactly with this line-of-nodes orientation,
+14
+
+Figure 7: The azimuthally-averaged surface-brightness profile of NGC 2655, according to the
+BASS survey data (taken from Legacy Survey, Dey et al. 2019), fitted by two exponential
+segments, µr = 19.4 + 1.086R′′/21.6′′, in the radius range of R = 26′′ − 50′′, and µr = 21.3 +
+1.086R′′/56.5′′, in the radius range of R = 70′′−120′′ (the left plot), and the LOS stellar velocity
+dispersion profile in the PA = 102◦ cross-section, according to the long-slit data (right).
+PA = 110◦, has given the intrinsic galaxy shape with the oval inner components; the Salo et
+al. (2015) team considered one of them as a triaxial bulge, and the other as a bar.
+We do not agree with this interpretation of the structure of NGC 2655. The fact is that
+the stellar LOS velocity field obtained for the central part of the galaxy with the SAURON IFU
+looks like a regular circular rotation (Dumas et al., 2007). We analyzed this velocity field using
+the tilted ring method and found the line-of-nods orientation for the stellar-component rotation
+plane PA = 263◦ ± 3◦ up to the distance of 25′′ from the center. Dumas et al. (2007) obtained
+with the kinemetry method PA = 266◦ ± 1◦ using the same data. The exact coincidence of the
+orientations of the photometric and kinematical major axes proves that the stars in the center
+of NGC 2655 rotate in circular orbits within the axisymmetric potential: the galaxy has no
+bar.
+Another important diagnostic feature of a thin stellar disk is that it must be dynamically
+cold: its rotation velocity must be several times greater than the stellar velocity dispersion.
+Figure 7, right, shows the profile of stellar velocity dispersion that we measured along the
+cross section with a long slit at PA = 102◦. The stellar line-of-sight velocities and velocity
+dispersions were measured by the cross-correlation method similar to that we used in the paper
+15
+
+dispersion,km/s
+200
+NGC2655PA=102
+150
+盂
+100
+W
+Stellar velocity
+50
+0
+-50
+0
+50
+r, arcsecFigure 8: The stellar LOS velocity profiles along two long-slit cross-sections close to the pho-
+tometric major axis: the red stars are for PA = 102◦, and the black stars – for PA = 115◦.
+by Silchenko et al. (2019). Already at the radius of R = 30′′, the stellar velocity dispersion
+drops to 50 km/s: this is the radial boundary, where the thin stellar disk begins to dominate.
+We have also shown in Fig. 7, left, the decomposition of the surface brightness profile consistent
+with the dominance of the disk so close to the center: the photometric disk of NGC 2655 has
+the type III profile, that is, it consists of two exponential segments, the inner one with a smaller
+scalelength than the outer (which was also found in the S4G survey).
+Thus, we can state that NGC 2655 has two exponential disks: they have different scale-
+lengths, but they also have different orientations of the isophote major axis. And along with
+this, the orientation of the major axis of the inner isophote is supported by the stellar kinemat-
+ics: as the analysis of the two-dimensional velocity field shows, this is indeed the line of nodes of
+the circular rotation plane. As for the outer disk, for which the orientation of the photometric
+major axis PA = 110◦ was found in the S4G survey, here we cannot properly compare it with
+the orientation of the kinematical major axis: there is no two-dimensional stellar velocity field
+over a large extension, R > 20′′, of the galaxy disk. But we have long-slit cross-sections in
+different slit orientations. Figure 8 compares the line-of-sight velocity profiles for the stellar
+component in the slit orientations PA = 102◦ and PA = 115◦. We can note that at the radius
+of R = 40′′ the rotation velocity projected onto the line of sight in PA = 115◦ is larger than
+that in PA = 102◦. This means that the kinematical major axis in the outer stellar disk is
+16
+
+closer to PA = 115◦ than to PA = 102◦ which excludes the line-of-nodes orientation found
+for the inner stellar component. At the same time, the photometric major-axis orientation of
+the outer disk may be the orientation of the line of nodes: our kinematical cross-sections with
+a long slit do not exclude this. It appears that the internal and external rotation axes of the
+stellar disk of NGC 2655 are inclined to each other, in other words, NGC 2655 is a multi-spin
+galaxy.
+5.2
+The Origin of Gas in NGC 2655
+The orientations of the huge disk of neutral hydrogen and the outer stellar disk in NGC 2655
+coincide with each other both spatially and kinematically. Previously, Sparke et al. (2008)
+noted that two billion solar masses of cold gas is too much for one minor merger, and several
+such events are needed (but with the same orientation of the infall orbits, because all the gas
+rotates in the same plane). Now we understand that these multiple minor mergers should have
+brought not only several billion solar masses of gas, but also several billion solar masses of stars
+for the outer stellar disk of NGC 2655, which makes the supposed multiple minor merging a
+quite incredible event. Opposite to Sparke et al. (2008), we conclude that the outer gaseous
+disk lies within the outer stellar disk, and even current star formation is taking place somewhere
+in it: it is at the southern edge of the disk that we have detected the gas emission excited by
+young stars, and the northern arc shows also an excess of ultraviolet (Fig. 4). The metallicity of
+the gas in this outer disk is solar, which is atypical for dwarf galaxies that Sparke et al. (2008)
+have suggested as the source of NGC 2655 gaseous disk. The entire external configuration of the
+galaxy resembles a classical large disk of a spiral galaxy, which, according to modern evolution
+concepts, is cumulated over billions years by smooth external accretion of cold gas (Tacconi et
+al., 2020), albeit from a source undefined still at a global scale.
+But minor merging certainly took place in NGC 2655. It also brought along a noticeable
+amount of the gas with the spin strongly decoupled from the regular rotation of the outer
+17
+
+disk, both stellar and gaseous. Apparently, a companion fell onto the galaxy almost vertically,
+and now, within two kiloparsecs from the center, we observe the remnants of the destroyed
+companion as a circumpolar loop – the picture is very similar to Sagittarius dwarf torn apart
+by the Milky Way. But in the case of NGC 2655, there was much more gas in the merged
+companion. The gas of the vertically infalling companion hit the gaseous disk of NGC 2655
+being in regular rotation, and this collision inevitably resulted in the development of shock
+fronts. The shock wave has not only excited the gas in the polar loop, it ran outward across
+the large galactic gaseous disk. At distances of up to 8 kpc from the center, we observe the gas
+of the large disk excited by this shock wave, although, the kinematics of this gas to the east
+of the nucleus is little affected and exhibits rotation consistent with that of the stellar disk. If
+the shock wave propagated at an average velocity of 1000 km/s , then the impact could have
+taken place approximately 107 years ago.
+6
+ACKNOWLEDGMENTS
+The paper is based on the observational data obtained with the 6-m telescope at the Special
+Astrophysical Observatory of the Russian Academy of Sciences (BTA SAO RAS) and with
+the 2.5-m telescope at the Caucasus Mountain Observatory of the Sternberg Astronomical
+Institute of the Moscow State University. The spectroscopic analysis was supported by the
+grant of the Russian Science Foundation no.22-12-00080, and the narrow-band photometry –
+by the grant of the Russian Fund for Basic Research no. 20-02-00080. The observations at the
+BTA SAO RAS telescope are supported by the Ministry of Science and Higher Education of the
+Russian Federation; the observational technique is improved in the frame of the National project
+”Science and universities”. In our analysis we used data from publicly accessible archives and
+databases: the Lyon-Meudon Extragalactic Database (LEDA) maintained by the LEDA team
+at the Lyon Observatory CRAL (France) and the NASA/IPAC Extragalactic Database (NED)
+operated by the Jet Propulsion Laboratory of the California Institute of Technology under
+18
+
+contract with the National Aeronautics and Space Administration (USA). We also invoked the
+data from the GALEX space telescope for our analysis. The NASA GALEX data were taken
+from the Mikulski Archive for Space Telescopes (MAST). In our figures we have also used the
+plots provided by observations made with the NASA/ESA Hubble Space Telescope and obtained
+from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science
+Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA)
+and the Canadian Astronomy Data Centre (CADC/NRC/CSA). The broad-band photometry
+is based on the data taken from the Legacy Survey resource (the BASS survey). The Legacy
+Surveys consist of three individual and complementary projects: the Dark Energy Camera
+Legacy Survey (DECaLS; Proposal ID no.2014B-0404; PIs: David Schlegel and Arjun Dey),
+the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID no.2015A-0801; PIs: Zhou Xu and
+Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID no.2016A-0453; PI:
+Arjun Dey). BASS is a key project of the Telescope Access Program (TAP), which has been
+funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences
+(the Strategic Priority Research Programs, Grant no. XDB09000000), and the Special Fund
+for Astronomy from the Ministry of Finance. The BASS is also supported by the External
+Cooperation Program of Chinese Academy of Sciences (Grant no. 114A11KYSB20160057),
+and Chinese National Natural Science Foundation (Grant no. 12120101003, no. 11433005).
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+

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+On Helly numbers of exponential lattices∗
+Gergely Ambrus1
+Martin Balko2
+N´ora Frankl3
+Attila Jung4
+M´arton Nasz´odi5
+1 Dpeartment of Geometry, Bolyai Institute, University of Szeged, Hungary, and
+Alfr´ed R´enyi Institute of Mathematics, Hungary
+[email protected]
+2 Department of Applied Mathematics,
+Faculty of Mathematics and Physics, Charles University, Czech Republic
+[email protected]
+3 School of Mathematics and Statistics, The Open University, UK, and Alfr´ed R´enyi Institute of
+Mathematics, Hungary
+[email protected]
+4 Institute of Mathematics, ELTE E¨otv¨os Lor´and University, Hungary
+[email protected]
+5 Alfr´ed R´enyi Institute of Mathematics and Department of Geometry, E¨otv¨os Lor´and University,
+Hungary.
+[email protected]
+Abstract
+Given a set S ⊆ R2, define the Helly number of S, denoted by H(S), as
+the smallest positive integer N, if it exists, for which the following statement
+is true: For any finite family F of convex sets in R2 such that the intersection
+of any N or fewer members of F contains at least one point of S, there is a
+point of S common to all members of F.
+∗G. Ambrus was partially supported by ERC Advanced Grant ”GeoScape”, by the Hungarian
+National Research grant no. NKFIH KKP-133819, and by project no. TKP2021-NVA-09, which
+has been implemented with the support provided by the Ministry of Innovation and Technology
+of Hungary from the National Research, Development and Innovation Fund, financed under the
+TKP2021-NVA funding scheme. M. Balko was supported by the grant no. 21/32817S of the
+Czech Science Foundation (GAˇCR) and by the Center for Foundations of Modern Computer
+Science (Charles University project UNCE/SCI/004). N. Frankl was partially supported by ERC
+Advanced Grant ”GeoScape”. A. Jung was supported by the R´enyi Doctoral Fellowship of the
+R´enyi Institute. M. Nasz´odi was supported by the J´anos Bolyai Scholarship of the Hungarian
+Acadamy of Sciences. This article is part of a project that has received funding from the European
+Research Council (ERC) under the European Union’s Horizon 2020 research and innovation
+programme (grant agreement No 810115).
+1
+arXiv:2301.04683v1  [math.CO]  11 Jan 2023
+
+We prove that the Helly numbers of exponential lattices {αn : n ∈ N0}2
+are finite for every α > 1 and we determine their exact values in some
+instances. In particular, we obtain H({2n : n ∈ N0}2) = 5, solving a problem
+posed by Dillon (2021).
+For real numbers α, β > 1, we also fully characterize exponential lattices
+L(α, β) = {αn : n ∈ N0} × {βn : n ∈ N0} with finite Helly numbers by
+showing that H(L(α, β)) is finite if and only if logα(β) is rational.
+1
+Introduction
+Helly’s theorem [9] is one of the most classical results in combinatorial geometry.
+It states that, for each d ∈ N, if the intersection of any d + 1 or fewer members of
+a finite family F of convex sets in Rd is nonempty, then the entire family F has
+nonempty intersection. There have been numerous variants and generalizations of
+this famous result; see [1, 11] for example. One active direction of this research
+with rich connections to the theory of optimization [1] is the study of variants of
+Helly’s theorem with coordinate restrictions, which is captured by the following
+definition.
+Let d be a positive integer. The Helly number of S of a set S ⊆ Rd, denoted
+by H(S), is the smallest positive integer N, if it exists, such that the following
+statement is true for every finite family F of convex sets in Rd: if the intersection of
+any N or fewer members of F contains at least one point of S, then � F contains
+at least one point of S. If no such number N exists, then we write H(S) = ∞.
+Helly’s theorem in this language can be restated as H(Rd) = d + 1.
+A classical result of this sort is Doignon’s theorem [6] where the set S is the
+integer lattice Zd. This result, which was also independently discovered by Bell [2]
+and by Scarf [13], states that H(Zd) ≤ 2d. This is tight as for Q = {0, 1}d the
+intersection of any 2d − 1 sets in the family {conv(Q \ {x}): x ∈ Q} contains a
+lattice point, but the intersection of all 2d sets does not.
+The theory of Helly numbers of general sets is developing quickly and there are
+many result of this kind [1, 11]. For example, De Loera, La Haye, Oliveros, and
+Rold´an-Pensado [3] and De Loera, La Haye, Rolnick, and Sober´on [4] studied the
+Helly numbers of differences of lattices and Garber [7] considered Hely numbers of
+crystals or cut-and-project sets.
+The Helly number of a set S is closely related to the maximum size of a set that
+is empty in S. A subset X ⊆ S is intersect-empty if
+��
+x∈X conv(X \ {x})
+�
+∩S = ∅.
+A convex polytope P with vertices in S is empty in S if P does not contain any
+points of S other than its vertices. For a discrete set S, we use h(S) to denote
+the maximum number of vertices of an empty polytope in S. If there are empty
+polytopes in S with arbitrarily large number of vertices, then we write h(S) = ∞.
+2
+
+The following result by Hoffman [10] (which was essentially already proved by
+Doignon [6]) shows the close connection between intersect-empty sets and empty
+polygons in S and the S-Helly numbers.
+Proposition 1 ([10]). If S ⊆ Rd, then H(S) is equal to the maximum cardinality
+of an intersect-empty set in S. If S is discrete, then H(S) = h(S).
+Since all the sets S studied in this paper are discrete, we state all of our results
+using h(α) but, due to Proposition 1, our results apply to H(α) as well.
+Very recently, Dillon [5] proved that the Helly number of a set S is infinite if S
+belongs to a certain collection of product sets, which are sets of the form S = Ad with
+a certain kind of discrete set A ⊆ R. His result shows, for example, that whenever
+p is a polynomial of degree at least 2 and d ≥ 2, then h({p(n): n ∈ N0}d) = ∞.
+However, there are sets for which Dillon’s method gives no information, for example
+{2n : n ∈ N0}2. Thus, Dillon [5] posed the following question, which motivated our
+research.
+Problem 1 (Dillon, [5]). What is h({2n : n ∈ N0}2)?
+In this paper, we study the Helly numbers of exponential lattices L(α) and
+L(α, β) in the plane where L(α) = {αn : n ∈ N0}2 and L(α, β) = {αn : n ∈
+N0} × {βn : n ∈ N0} for real numbers α, β > 1. In particular, we prove that Helly
+numbers of exponential lattices L(α) are finite and we provide several estimates
+that give exact values for α sufficiently large, solving Problem 1. We also show
+that Helly numbers of exponential lattices L(α, β) are finite if and only if logα(β)
+is rational.
+2
+Our results
+For a real number α > 1 and the exponential lattice L(α) = {αn : n ∈ N0}2, we
+abbreviate h(L(α)) by h(α).
+As our first result, we provide finite bounds on the numbers h(α) for any α > 1.
+The upper bounds are getting smaller as α increases and reach their minimum at
+α = 2.
+Theorem 2. For every real α > 1, the maximum number of vertices of an empty
+polygon in L(α) is finite. More precisely, we have h(α) ≤ 5 for every α ≥ 2,
+h(α) ≤ 7 for every α ≥ [ 1+
+√
+5
+2
+, 2), and
+h(α) ≤ 3
+�
+logα
+�
+α
+α − 1
+��
++ 3
+for every α ∈ (1, 1+
+√
+5
+2
+).
+3
+
+We note that if α = 1 + 1
+x for x ∈ (0, ∞), then the bound from Theorem 2
+becomes h(1 + 1
+x) ≤ O(x log2(x)). Moreover, we show that the breaking points
+of α for our upper bounds are determined by certain polynomial equations; see
+Section 3.
+We also consider the lower bounds on h(α) and provide the following estimate.
+Theorem 3. We have h(α) ≥ 5 for every α ≥ 2 and h(α) ≥ 7 for every α ∈
+�
+1+
+√
+5
+2
+, 2
+�
+. For every α ∈
+�
+1, 1+
+√
+5
+2
+�
+, we have
+h(α) ≥
+��
+1
+α − 1
+�
+.
+If α = 1 + 1
+x where x ∈ (0, ∞), then the lower bound from Theorem 3 becomes
+h(1 + 1
+x) ≥ ⌊√x⌋. So with decreasing α, the parameter h(α) indeed grows to
+infinity.
+By combining Theorems 2 and 3, we get the precise value of the Helly numbers
+of L(α) with α ≥ (1 +
+√
+5)/2. In particular, for α = 2, we obtain a solution to
+Problem 1.
+Corollary 4. We have h(α) = 5 for every α ≥ 2 and h(α) = 7 for every α ∈
+[ 1+
+√
+5
+2
+, 2).
+We prove the following result which shows that even a slight perturbation of S
+can affect the value h(S) drastically (note that this also follows by adding large
+empty polygons to S without changing its asymptotic density). We use the Fibonacci
+numbers (Fn)n∈N0, which are defined as F0 = 1, F1 = 1 and Fn = Fn−1 + Fn−2 for
+every integer n ≥ 2.
+Proposition 5. We have h({Fn : n ∈ N0}2) = ∞.
+We recall that Fn = ϕn+1−ψn+1
+√
+5
+for every n ∈ N0, where ϕ = 1+
+√
+5
+2
+is the golden
+ratio and ψ = 1−
+√
+5
+2
+= 1 − ϕ is its conjugate. Since ψ < 1, this formula shows that
+the points of {Fn : n ∈ N0}2 are approaching the points of the scaled exponential
+lattice
+ϕ
+√
+5 · L(ϕ) = { ϕ
+√
+5 · ϕn : n ∈ N0}2. Thus, Proposition 5 is in sharp contrast
+with the fact that h( ϕ
+√
+5 · L(ϕ)) = h(ϕ) ≤ 7, which follows from Theorem 2 and
+from the fact that affine transformations of any set S ⊆ Rd do not change h(S).
+We also note that the set {Fn : n ∈ N0}2 is not a product set for which Dillon’s
+method [5] gives h({Fn : n ∈ N0}2) = ∞.
+We also consider the more general case of exponential lattices where the rows
+and the columns might use different bases. For real numbers α > 1 and β > 1, let
+L(α, β) be the set {αn : n ∈ N0} × {βn : n ∈ N0}. Note that L(α) = L(α, α) for
+every α > 1.
+4
+
+As our last main result, we fully characterize exponential lattices L(α, β) with
+finite Helly numbers h(L(α, β)), settling the question of finiteness of Helly numbers
+of planar exponential lattices completely.
+Theorem 6. Let α > 1 and β > 1 be real numbers. Then, h(L(α, β)) is finite if
+and only if logα(β) is a rational number.
+Moreover, if logα(β) ∈ Q, that is, β = αp/q for some p, q ∈ N, then
+�
+1
+pq
+��
+1
+α1/q − 1
+��
+≤ h(L(α, β)) ≤ pq · h(αp) + 1.
+The proof of Theorem 6 is based on Theorem 2 and on the theory of continued
+fractions and Diophantine approximations.
+Open problems
+First, it is natural to try to close the gap between the upper bound from Theorem 2
+and the lower bound from Theorem 3 and potentially obtain new precise values of
+h(α).
+Second, we considered only the exponential lattice in the plane, but it would be
+interesting to obtain some estimates on the Helly numbers of exponential lattices
+{αn : n ∈ N0}d in dimension d > 2.
+We also mention the following conjecture of De Loera, La Haye, Oliveros, and
+Rold´an-Pensado [3], which inspired the research of Dillon [5].
+Conjecture 1 ([3]). If P is the set of prime numbers, then h(P2) = ∞.
+Using computer search, Summers [14] showed that h(P2) ≥ 14.
+3
+Proof of Theorem 2
+Here, we prove Theorem 2 by showing that the number h(α) is finite for every
+α > 1. This follows from the upper bounds h(α) ≤ 5 for α ≥ 2, h(α) ≤ 7 for every
+α ≥ [ 1+
+√
+5
+2
+, 2), and
+h(α) ≤ 3
+�
+logα
+�
+α
+α − 1
+��
++ 3
+for any α ∈ (1, 1+
+√
+5
+2
+).
+We start by introducing some auxiliary definitions and notation. Let α > 1
+be a real number and consider the exponential lattice L(α). For i ∈ N0, the ith
+column of L(α) is the set {(αi, αn): n ∈ N0}. Analogously, the ith row of L(α) is
+the set {(αn, αi): n ∈ N0}.
+5
+
+For a point p in the plane, we write x(p) and y(p) for the x- and y-coordinates
+of p, respectively. Let P be an empty convex polygon in L(α). Let e be an edge
+of P connecting vertices u and v where x(u) < x(v) or y(u) < y(v) if x(u) = x(v).
+We use e to denote the line determined by e and oriented from u to v. The slope
+of e is the slope of e, that is, y(v)−y(u)
+x(v)−x(u).
+We distinguish four types of edges of P; see part (a) of Figure 1. First, assume
+x(u) ̸= x(v) and y(u) ̸= y(v). We say that e is of type I if the slope of e is negative
+and P lies to the right of e. Similarly, e is of type II if the slope of e is positive
+and P lies to the right of e. An edge e has type III if the slope of e is negative
+and P lies to the left of e. Finally, type IV is for e with positive slope and with
+P lying to the left of e. It remains to deal with horizontal and vertical edges of
+P. A horizontal edge e is of type II if P lies below e and is of type III otherwise.
+Similarly, a vertical edge e is of type IV if P lies to the left of e and is of type III
+otherwise.
+(a)
+(b)
+I
+II
+III
+IV
+0
+u = (αk, αℓ)
+v = (αk+m, αℓ−n)
+(αk+m+r, 0)
+�
+��
+�
+≤ r − 1
+Figure 1: (a) The four types of edges of a convex polygon. (b) An illustration of
+the proof of Lemma 7.
+Note that each edge of P has exactly one type and that the types partition the
+edges of P into four convex chains. We first provide an upper bound on the number
+of edges of those chains of P and then derive the bound on the total number of
+edges of P by summing the four bounds. We start by estimating the number of
+edges of P of type I.
+Lemma 7. The polygon P has at most
+�
+logα
+�
+α
+α−1
+��
+edges of type I.
+Proof. First, let r =
+�
+logα
+�
+α
+α−1
+��
+and note that r ≥ 1 as α > 1. Let e be the
+left-most edge of P of type I and let u and v be vertices of e. Since e is of type I,
+we have u = (αk, αℓ) and v = (αk+m, αℓ−n) for some positive integers k, ℓ, m, and
+n.
+We will show that the point (αk+m+r, 0) lies above the line e. Since there are at
+most r − 1 columns of L(α) between the vertical line containing v and the vertical
+line containing (αk+m+r, 0) and the point (αk+m+r, 0) is below the lowest row of
+6
+
+L(α), it then follows that there are at most r edges of P of type I; see part (b) of
+Figure 1.
+Since the line e contains u and v, we see that
+e = {(x, y) ∈ R2 : (αℓ − αℓ−n)x + (αk+m − αk)y = αk+ℓ+m − αk+ℓ−n}.
+It suffices to check that by substituting the coordinates of the point (αk+m+r, 0)
+into the equation of the line e gives a left side that is at least αk+ℓ+m − αk+ℓ−n.
+The left side equals αk+ℓ+m+r − αk+ℓ+m−n+r and thus we want
+αk+ℓ+m+r − αk+ℓ+m−n+r ≥ αk+ℓ+m − αk+ℓ−n.
+By dividing both sides by αk+ℓ and by rearranging the terms, we can rewrite this
+expression as
+α−n(1 − αm+r) ≥ αm − αm+r.
+Since m, r > 0 and α > 1, we get (1−αm+r) < 0 and thus the left side is increasing
+as n increases, so we can assume n = 1, leading to
+α−1 − αm+r−1 ≥ αm − αm+r.
+We can again rearrange the inequality as
+αr − αr−1 − 1 ≥ −α−1−m,
+where the right side is negative and approaches 0 as m tends to infinity, so we can
+replace it by 0, obtaining
+αr − αr−1 ≥ 1.
+This inequality is satisfied by our choice of r.
+We now estimate the number of edges of P that are of type III.
+Lemma 8. The polygon P has at most 2⌈logα
+� α+1
+α
+�
+⌉ + 1 edges of type III for
+1 < α < 2 and at most 2 such edges for α ≥ 2.
+Proof. Let t = ⌈logα
+� α+1
+α
+�
+⌉ and s = t + 1 for α ∈ (1, 2) and t = 1 = s for α ≥ 2.
+Suppose for contradiction that there are s + t + 1 edges of P of type III. Let
+v1, . . . , vs+t+2 be the vertices of the convex chain that is formed by edges of P of
+type III. We use Q to denote the convex polygon with vertices v1, . . . , vs+t+2. Note
+that Q is empty in L(α) as P is empty and Q ⊆ P.
+Let v′ be the point (x(vs+2), α · y(vs+2)), that is, v′ is the point of L(α) that
+lies just above vs+2; see part (a) of Figure 2. We will show that the point v′ lies
+below the line v1vs+t+2. Since v′ lies in the same column of L(α) as vs+2, this then
+7
+
+(b)
+0
+u
+v
+v′
+W
+(a)
+0
+x(vs+t+2)
+αt
+v′
+vs+t+2
+v1
+v2
+v3
+y(v1)
+αs
+Q
+v′′
+Figure 2: (a) An illustration of the proof of Lemma 8 for s = 1 = t. (b) An
+illustration of the proof of Lemma 9.
+implies that v′ lies in the interior of Q, contradicting the fact that Q is empty in
+L(α).
+Note that x(v′) ≤ x(vs+t+2)
+αt
+and y(v′) ≤ y(v1)
+αs
+as all edges vivi+1 are of type III
+and thus the x- and y-coordinates decrease by a multiplicative factor at least α for
+each such edge. Since the only vertical edge might be v1v2 and the only horizontal
+edge might be vs+t+1vs+t+2, the x- or y-coordinates indeed decrease by the factor
+α at each step.
+Let v1 = (αk, αℓ) and vs+t+2 = (αk+m, αℓ−n) for some positive integers k, ℓ, mn, n.
+Note that m, n ≥ s + t. The line determined by v1 and vs+t+2 is then
+{(x, y) ∈ R2 : (αℓ − αℓ−n)x + (αk+m − αk)y = αk+ℓ+m − αk+ℓ−n}.
+Since x(v′) ≤ x(vs+t+2)
+αt
+and y(v′) ≤ y(v1)
+αs , it suffices to check
+(αℓ − αℓ−n)αk+m
+αt
++ (αk+m − αk)αℓ
+αs < αk+ℓ+m − αk+ℓ−n.
+After dividing by αk+ℓ+m, this can be rewritten as
+α−t + α−s < 1 − α−m−n + α−t−n + α−s−m.
+Since m, n ≥ s + t, the right hand side is decreasing with increasing m and n and
+thus we only need to prove
+α−s + α−t ≤ 1.
+If α ≥ 2, then s = 1 = t and this inequality becomes 2/α ≤ 1, which is true. If
+α ∈ (1, 2), then s = t + 1 and the inequality becomes 1 + 1/α ≤ αt, which is also
+true by our choice of t.
+It remains to bound the number of edges of P that are of types II and IV.
+Observe that if we switch the x- and y- coordinates of P, then edges of type II
+8
+
+become edges of type IV and vice versa. Since the exponential lattice L(α) is
+symmetric with respect to the line x = y, we see that it suffices to estimate the
+number of edges of type II. To do so, we use the following auxiliary result.
+Lemma 9. Let u be a point of L(α) and let v and v′ be two points of L(α) that are
+consecutive in a row R of L(α) that lies above the row containing u; see part (b)
+of Figure 2. Then, all points of L(α) that lie above R in the interior of the wedge
+spanned by the lines uv and uv′ lie on at most
+�
+logα(
+α
+α−1)
+�
+lines containing the
+origin.
+Proof. Similarly as in Lemma 7, we set r =
+�
+logα
+�
+α
+α−1
+��
+and note that r ≥ 1. We
+can assume without loss of generality that u = (1, 1) as otherwise it suffices to
+renumber the points of L(α). We can also assume without loss of generality that
+neither of the points v and v′ lies above the line x = y as v and v′ are consecutive
+on R and thus both cannot lie in opposite open halfplanes determined by this line.
+Let o be the origin and consider the lines ov and ov′. Then, the part of the line
+ov above the row R is above uv; see part (b) of Figure 2. Similarly, the part of the
+line ov′ above R is above uv′. It follows that only points of L(α) that lie on a line
+ow, where w is a point of L(α) to the right of v on R, can lie in the interior of W.
+Let v′′ be the point (αr · x(v′), y(v′)), that is, v′′ is the point of L(α) that lies
+at distance r to the right of v′ on R, We will show that the part of the line ov′′
+above R lies below uv′. This will conclude the proof as all points of L(α) that lie
+in the interior of W above R have to then lie on one of the r lines ow with w lying
+between v and v′′ on R.
+It suffices to compare the slopes of the lines ov′′ and uv′. Let v′ = (αm, αn) for
+some positive integers m and n. Then, the slope of ov′′ is
+y(v′′) − y(o)
+x(v′′) − x(o) =
+y(v′)
+αr · x(v′) =
+αn
+αm+r
+and the slope of uv′ equals
+y(v′) − y(u)
+x(v′) − x(u) = y(v′) − 1
+x(v′) − 1 = αn − 1
+αm − 1.
+Thus, we want
+αn − 1
+αm − 1 ≥
+αn
+αm+r .
+We can rewrite this inequality as
+αm+n+r − αm+r ≥ αn+m − αn,
+which can be further rewritten by dividing both sides with αn as
+αm+r(1 − α−n) ≥ αm − 1.
+9
+
+The left side is increasing with increasing n, so we can assume n = 1 and, by
+dividing both sides with αm, we obtain
+αr(1 − α−1) ≥ 1 − α−m.
+Now, the term α−m on the right side approaches 0 from below with increasing m,
+so we can replace it by 0 obtaining
+αr − αr−1 ≥ 1.
+This inequality is satisfied by our choice of r.
+Now, we can apply Lemma 9 to obtain an upper bound on the number of edges
+of P of type II.
+Lemma 10. The polygon P has at most
+�
+logα
+�
+α
+α−1
+��
++ 1 edges of type II.
+Proof. Again, let r =
+�
+logα
+�
+α
+α−1
+��
+. Let u be the leftmost vertex of the convex
+chain C determined by the edges of P of type II. Similarly, let v be the second
+leftmost vertex of C. Note that since the edge uv is of type II, the vertex v lies in
+a row R of L(α) above the row containing u. Let v′ be the point (α · x(v), y(v)),
+that is, point of L(α) that is to the right of v on R. Then, by Lemma 9, all points
+of L(α) that lie above R and in the interior of the wedge W spanned by the lines
+uv and uv′ lie on at most r lines containing the origin.
+Since P is empty in L(α), all vertices of C besides u, v, and possibly v′ lie in
+W above R. Since all edges of C are of type II, every line determined by the origin
+and by a point of L(α) from the interior of W contains at most one vertex of C.
+Note that if v′ is a vertex of C, then the only vertices of C are u, v, v′. Thus, in
+total C has at most r + 2 vertices and therefore at most r + 1 edges.
+We recall that, by symmetry, the same bound applies for edges of type IV and
+thus we get the following result.
+Corollary 11. The polygon P has at most
+�
+logα
+�
+α
+α−1
+��
++ 1 edges of type IV.
+Since each edge of P is of one of the types I–IV, it immediately follows from
+Lemmas 7, 8, 10, and from Corollary 11 that the number of edges of P is at most
+3
+�
+logα
+�
+α
+α − 1
+��
++ 2 + 2
+�
+logα
+�α + 1
+α
+��
++ 1 ≤ 5
+�
+logα
+�
+α
+α − 1
+��
++ 3,
+as logx
+�
+x
+x−1
+�
+≥ logx
+� x+1
+x
+�
+for every x > 1. In particular, this gives h(2) ≤ 8 and
+h
+�
+1+
+√
+5
+2
+�
+≤ 13. To obtain better bounds that are tight for α ≥ 1+
+√
+5
+2
+, we observe
+10
+
+that not all types can appear simultaneously. To show this, we will use one last
+auxiliary result.
+Let p and q be (not necessarily different) points lying on the same row R of
+R(α), each contained in an edge of P. Let L and L′ be two lines containing p and
+q, respectively. If the slopes of L and L′ are negative, then we call the part of the
+plane between L and L′ below R a slice of negative slope; see part (a) of Figure 3
+Analogously, a slice of positive slope is the part of the plane between L and L′
+above R if L and L′ have positive slope.
+(a)
+(b)
+0
+P
+q
+p
+L′
+L
+R
+0
+P
+q
+p
+L′
+L
+R
+v
+w
+u
+Figure 3: (a) An example of a slice of negative slope. The slice is denoted by dark
+gray stripes. (b) An illustration of the proof of Lemma 12 for negative slopes.
+Lemma 12. If the polygon P is contained in a slice of negative slope, then there
+is no non-vertical edge of P of type IV. Similarly, if P is contained in a slice of
+positive slope, then there is no edge of type I.
+Proof. By symmetry, it suffices to prove the statement for slices of negative slope.
+Suppose for contradiction that there is a non-vertical edge uv of type IV in a slice of
+negative slope determined by lines L and L′ and points p and q as in the definition
+of a slice. Without loss of generality, we assume x(u) < x(v).
+Consider the point w = (x(u), y(v)) of L(α). Since uv is non-vertical, we have
+w /∈ {u, v}. We claim that w is in the interior of P, contradicting the assumption
+that P is empty in L(α). Since uv is of type IV, the point u lies below the row
+containing w. However, since p is contained in an edge of P and P is in the slice,
+the boundary of P intersects this row to the left of w. Analogously, v is to the right
+of the column containing w and thus the boundary of P intersects this column
+above w. Then, however, w lies in the interior of P.
+Finally, we can now finish the proof of Theorem 2.
+Proof of Theorem 2. First, we observe that if all vertices of P lie on two columns
+of L(α), then P can have at most four vertices. So we assume that this is not the
+11
+
+case. Let u be the leftmost vertex of P with the highest y-coordinate among all
+leftmost vertices of P. Let e1 and e2 be the edges of P incident to u. We denote
+the other edge of P incident to e1 as e. We also use tI, tII, tIII, and tIV to denote
+the number of edges of P of type I, II, III, and IV, respectively.
+First, assume that e1 is vertical. If e2 is horizontal, then, since u is the top vertex
+of e1 and P is not contained in two columns of L(α), the point (α · x(u), y(u)/α)
+of L(α) lies in the interior of P, which is impossible as P is empty in L(α).
+(a)
+(b)
+(c)
+(d)
+e1
+u
+R
+e2
+e
+e1
+u
+R
+e2
+e
+e1
+u
+R
+e2
+e2
+u
+e1
+u
+R
+e
+Figure 4: An illustration of the proof of Theorem 2.
+If e1 is vertical and the slope of e2 is negative, then there is no edge of type
+II. Thus, the edge e intersects the row R of L(α) containing the other vertex of e1
+and e has negative slope. Then, the part of P below R is contained in the slice
+of negative slope determined by e2 and e; see part (a) of Figure 4. By Lemma 12,
+there is no non-vertical edge of type IV in P. By Lemmas 7 and 8, the total number
+of edges of P is thus at most
+tI + tIII + 1 ≤
+�
+logα
+�
+α
+α − 1
+��
++ 2
+�
+logα
+�α + 1
+α
+��
++ 1
+for α ∈ (1, 2) and is by one smaller for α ≥ 2.
+If e1 is vertical and the slope of e2 is positive, then, since P is empty, there is
+no edge of type III besides e1 as otherwise the point (α · x(u), y(u)) of L(α) is in
+the interior of P. The edge e intersects the row R of L(α) containing u and e has
+positive slope. Thus, the part of P above R is contained in the slice of positive
+slope determined by e2 and e; see part (b) of Figure 4. By Lemma 12, there is no
+edge of type I in P. By Lemma 10 and Corollary 11, the total number of edges of
+P is then at most
+tII + 1 + tIV ≤ 2
+�
+logα
+�
+α
+α − 1
+��
++ 3.
+In the rest of the proof, we can now assume that none of the edges e1 and e2 is
+vertical. We can label them so that the slope of e1 is larger than the slope of e2.
+12
+
+First, assume that the slope of e1 is positive and the slope of e2 is negative. Then,
+since the vertices of P do not lie on two columns of L(α), the point (α · x(u), y(u))
+is contained in the interior of P, which is impossible as P is empty in L(α).
+If the slopes of e1 and e2 are both non-positive, then there is no edge of type
+II besides the possibly horizontal edge e1 as u is the leftmost vertex of P. By
+Lemma 12, there is also no non-vertical edge of type IV as P is contained in the
+slice of negative slopes determined by e1 and e2 or by e and e2 if e1 is horizontal;
+see part (c) of Figure 4. Thus, by Lemmas 7 and 8, the number of edges of P is at
+most
+tI + 1 + tIII + 1 ≤
+�
+logα
+�
+α
+α − 1
+��
++ 2
+�
+logα
+�α + 1
+α
+��
++ 3
+for α ∈ (1, 2) and is by one smaller for α ≥ 2.
+If the slopes of e1 and e2 are both non-negative, then there is no edge of type
+III besides the possibly horizontal edge e2 (note that a vertical edge of type III
+would have u as its bottom vertex, which is impossible by the choice of u). Then,
+P is contained in the slice of positive slope determined by e1 and e2 or, if e2 is
+horizontal, by e1 and e; see part (d) of Figure 4. Lemma 12 then implies that there
+is also no edge of type I. We thus have at most
+tII + 1 + tIV ≤ 2
+�
+logα
+�
+α
+α − 1
+��
++ 3
+edges of P by Lemma 10 and Corollary 11.
+Altogether, the upper bound on the number of edges of P is
+max
+��
+logα
+�
+α
+α − 1
+��
++ 2
+�
+logα
+�α + 1
+α
+��
++ 3, 2
+�
+logα
+�
+α
+α − 1
+��
++ 3
+�
+for α ∈ (1, 2) and the first term is smaller by 1 for α ≥ 2. This becomes 5 for
+α ≥ 2, h(α) ≤ 7 for α ≥ [ 1+
+√
+5
+2
+, 2), and at most 3
+�
+logα
+�
+α
+α−1
+��
++ 3 otherwise, since
+�
+logα
+� α+1
+α
+��
+≤
+�
+logα
+�
+α
+α−1
+��
+for every α ∈ (1, 1+
+√
+5
+2
+).
+4
+Proof of Theorem 3
+We prove the lower bounds on h(α) through the following three propositions.
+Proposition 13. For every α ≥ 2, we have h(α) ≥ 5.
+Proof. It is easy to check that conv{(1, α2), (α, α), (α2, 1), (α2, α), (α, α2)} is an
+empty polygon in L(α) with 5 vertices for any α.
+Proposition 14. For every α ∈ [ 1+
+√
+5
+2
+, 2), we have h(α) ≥ 7.
+13
+
+Proof. Let k = k(α) be a sufficiently large integer, and let
+Q(α) = {(1, αk), (αk−2, αk−1), (αk−1, αk−2), (αk, 1), (αk, α), (αk−1, αk−1), (α, αk)};
+see Figure 5. We will show that conv(Q(α)) is an empty polygon in L(α) with 7
+vertices.
+Q(α)
+Figure 5: An illustration of the proof of Proposition 14.
+First, we show that Q(α) \ {(αk−1, αk−1)} is in convex position. For this, by
+symmetry, it is enough to check that the vector (αk−1, αk−2) − (αk, 1) is to the
+left of (1, αk) − (αk, 1). This is the case exactly if αk−1 − αk + αk−2 − 1 < 0. By
+rearranging we get αk−2(α+1−α2) < 1, which holds for any k, since α+1−α2 ≤ 0
+as α ≥ (1 +
+√
+5)/2.
+Now, to show that the set Q(α) is in convex position, it is sufficient to check
+that (αk−1, αk−1) − (αk, α) is to the left of (1, αk) − (αk, α). This holds exactly
+if αk−1 − αk + αk−1 − α ≥ 0. By rearranging we get 2αk−2(2 − α) ≥ 1. Since
+1 < α < 2, this holds if k is sufficiently large.
+Thus, conv(Q(α)) has 7 vertices. To show that conv(Q(α)) is empty in L(α),
+we remark that points of the exponential lattice L(α) with at least one coordinate
+smaller than αk−1 are below the line through (αk−1, αk−2) and (αk−2, αk−1). Further,
+points with at least one coordinate larger than αk−1 are either above the line through
+(1, αk) and (α, αk) or to the right of the line through (αk, 1) and (αk, α).
+Proposition 15. For every α > 1, we have h(α) ≥
+��
+1
+α−1
+�
+.
+Proof. For a positive integer k, let P(k) = {(αi, αk−i) : 1 �� i ≤ k}. Since P(k)
+is contained in the hyperbola h = {(x, y) ∈ R2 : x, y > 0, xy = αk}, the points of
+P(k) are in convex position, and conv(P(k)) has k vertices. We will show that if
+k ≤
+�
+1
+α−1, then conv(P(k)) is empty.
+For points (x, y) of L(α) above h, we have xy ≥ αk+1. Further, points (x, y)
+of L(α) with xy ≥ αk+2 are separated from h by the hyperbola h′ = {(x, y) ∈
+14
+
+R2 : x, y > 0, xy = αk+1}. Thus, it is sufficient to check that h′ is above the line ℓ
+connecting (1, αk) with (αk, 1). The closest point of h′ to ℓ is (α(k+1)/2, α(k+1)/2), thus
+it is sufficient to check that this point is above ℓ. This holds if 2α(k+1)/2−αk −1 ≥ 0
+and we show that this inequality is satisfied for k ≤
+�
+1
+α−1.
+Let α = 1 + s2 with some s ∈ (0, 1). In this notation, k ≤ 1/s and we need to
+prove that 2(1 + s2)(k+1)/2 ≥ (1 + s2)k + 1. Since (1 + s2)(k+1)/2 ≥ 1 + s2 k+1
+2
+by
+the Bernoulli inequality, and (1 + s2)k ≤ es2k, it is sufficient to prove the stronger
+inequality 2(1 + s2 k+1
+2 ) ≥ es2k + 1. The worst case, when k = 1/s, is equivalent to
+1 + s + s2 ≥ es, which holds for s ∈ (0, 1) as can be seen by the Taylor expansion
+of es.
+5
+Proof of Proposition 5
+Let us denote F = {Fn : n ∈ N0}2. Suppose for contradiction that there is a
+positive integer k such that h(F) ≤ k. We will show that the points (Fi+2, Fi)
+with odd i ∈ {1, . . . , 2k + 1} are vertices of an empty convex polygon P in F,
+contradicting the assumption h(F) ≤ k.
+First, we show that the points (Fi+2, Fi) with odd i ∈ {1, . . . , 2k + 1} are in
+convex position. to show that, it suffices to show that the slopes of lines determined
+by three consecutive such points are decreasing. That is, we want to prove
+Fi − Fi−2
+Fi+2 − Fi
+>
+Fi+2 − Fi
+Fi+4 − Fi+2
+for every odd i ∈ {1, . . . , 2k − 3}. Since Fk = Fk−1 + Fk−2 for every k ≥ 2, this
+inequality can be rewritten as
+Fi−1
+Fi+1
+> Fi+1
+Fi+3
+.
+Thus, we want to show that Fi−1 · Fi+3 > F 2
+i+1.
+Again, using the Fibonacci
+recurrence, we can rewrite this expression as Fi−1(3Fi+2Fi−1) > F 2
+i +2Fi−1·Fi+F 2
+i−1,
+which can be simplified to Fi−1(Fi + Fi−1) > F 2
+i and further to Fi−1 · Fi+1 > F 2
+i .
+Using the Binet formula Fk = ϕk+1−ψk+1
+√
+5
+, we see that this inequality is equivalent
+with
+(ϕi − ψi)(ϕi+2 − ψi+2) > (ϕi+1 − ψi+1)2.
+This can be expanded and rearranged to
+2ϕi+1 · ψi+1 > ϕi+2 · ψi + ϕi · ψi+2.
+15
+
+Since i is odd, ψi+1 is positive, and by diving both sides by ϕi+1 · ψi+1, we obtain
+2 > ϕ
+ψ + ψ
+ϕ = −3,
+as ϕ = 1+
+√
+5
+2
+and ψ = 1−
+√
+5
+2
+. Thus, the points are indeed in convex position.
+To show that the polygon P is empty in F, consider the line L = {(x, y) ∈
+R2 : y = x/ϕ2}. Any point (Fi+2, Fi) with odd i lies below L because
+Fi+2
+ϕ2
+= 1
+ϕ2 · ϕi+3 − ψi+3
+√
+5
+> ϕi+1 − ψi+1
+√
+5
+= Fi
+since ϕ2 > ψ2 and i + 3, i + 1 are both even implying ψi+3, ψi+1 > 0. Analogously,
+all points (Fi+2, Fi) with even i lie above L. For any i, every point (Fj, Fi) with
+j ≤ i + 1 lies above L, because Fi ≥ Fj−1 > Fj/ϕ2. Each point (Fi+2, Fi) with
+i > n lies at vertical distance less than 1/2 from L as
+Fi+2
+ϕ2
+= 1
+ϕ2 · ϕi+3 − ψi+3
+√
+5
+= ϕi+1 − ψi+1
+√
+5
++ ϕ2ψi+1 − ψi+3
+ϕ2√
+5
+≤ Fi + ϕ2ψ2 − ψ4
+√
+5
+< Fi + 1
+2.
+Any point (Fi+2, Fj) with j ≤ i − 1 lies below L at vertical distance at least
+1/2 since the distance is either at least Fi − Fj ≥ 1 if i is odd or it is at least
+Fi − Fj − 1
+2 ≥ 1
+2 if i is even. Thus, all points from F \ P are either above L or lie
+at vertical distance at least 1/2 from L. It follows that P is empty convex polygon
+in F and h(F) ≥ k + 1, a contradiction.
+6
+Proof of Theorem 6
+Let α, β > 1 be two real numbers. We prove that h(L(α, β)) is finite if and only if
+logα(β) is a rational number.
+6.1
+Finite upper bound
+First, assume that logα(β) ∈ Q. We will use Theorem 2 to show that the number
+h(L(α, β)) is finite. Since logα(β) ∈ Q and α, β > 1, there are positive integers
+p and q such that β = αp/q. Suppose for contradiction that there is an empty
+polygon P in L(α, β) with at least pq · h(αp) + 1 vertices. Note that this number
+of vertices is finite by Theorem 2. For k ∈ {0, . . . , q − 1}, we call a row of L(α, β)
+congruent to k if it is of the form R×βm for some integer m congruent to k modulo
+16
+
+q. Analogously, a column of L(α, β) is congruent to ℓ ∈ {0, . . . , p − 1} if it is of the
+form αm × R for some m congruent to ℓ modulo p.
+Now, since P contains at least pq · h(αp) + 1 vertices, the pigeonhole principle
+implies that there are integers k ∈ {0, . . . , q − 1} and ℓ ∈ {0, . . . , p − 1} such that
+at least h(αp) + 1 vertices of P that all lie in rows congruent to k and in columns
+congruent to ℓ. Let P ′ be the convex polygon that is spanned by these vertices.
+We claim that the polygon P ′ is not empty in L(α, β). Since P ′ ⊆ P, we get that
+P is also not empty in L(α, β), which contradicts our assumption about P.
+To show that P ′ is not empty in L(α, β), consider the subset L of L(α, β) that
+contains only points of L(α, β) that lie in rows congruent to k and in columns
+congruent to ℓ. Clearly, vertices of P ′ lie in L and L is an affine image of L(αp),
+which is scaled by the factors αℓ and αk in the x- and y-direction, respectively.
+Since affine mappings preserve incidences and P ′ has at least h(αp) + 1 vertices, it
+follows that P ′ is not empty in L. Since L ⊆ L(α, β), P ′ is not empty in L(α, β)
+either.
+6.2
+Finite lower bound
+Let logα(β) ∈ Q and β = αp/q for some relative prime positive integers p and q.
+Observe that in this case L(α, β) ⊂ L(α1/q). Thus, if an empty polygon in L(α1/q)
+is a subset of L(α, β), then it is an empty polygon in L(α, β).
+Let k =
+��
+1/(α1/q − 1)
+�
+and consider the set P = {(αi/q, α(k−i)/q) : 1 ≤ i ≤ k}.
+It is an empty polygon in L(α1/q), as it is shown in the proof of Proposition 15.
+Since its subset P ′ = {(αi/q, α(k−i)/q) : 1 ≤ i ≤ k with q|i and p|k − i} is a subset
+of L(α, β) and an empty polygon in L(α1/q), it is an empty polygon in L(α, β) with
+⌊k/pq⌋ vertices.
+6.3
+Infinite lower bound
+Now, assume that logα(β) /∈ Q. We will find a subset of L(α, β) forming empty
+convex polygon in L(α, β) with arbitrarily many vertices. To do so, we use a theory
+of continued fractions, so we first introduce some definitions and notation.
+6.3.1
+Continued fractions
+Here, we recall mostly basic facts about so-called continued fractions, which we
+use in the proof. Most of the results that we state can be found, for example, in
+the book by Khinchin [12].
+17
+
+For a positive real number r, the (simple) continued fraction of r is an expression
+of the form
+r = a0 +
+1
+a1 +
+1
+a2+
+1
+a3+···
+,
+where a0 ∈ N0 and a1, a2, . . . are positive integers. The simple continued fraction
+of r can be written in a compact notation as
+[a0; a1, a2, a3, . . . ].
+For every n ∈ N0, if we denote pn
+qn = [a0; a1, a2, . . . , an] and set p−1 = 1, p0 = a0,
+q−1 = 0, q0 = 1, then the numbers pn and qn satisfy the recurrence
+pn = anpn−1 + pn−2
+and
+qn = anqn−1 + qn−2
+(1)
+for each n ∈ N. Observe that if r is irrational, then its continued fraction has
+infinitely many coefficients. Also, it follows from (1) that pn
+qn < r for n even and
+pn
+qn > r for n odd.
+For example, if r = log2(3), we get the continued fraction [1; 1, 1, 2, 2, 3, 1, 5, 2, 23, . . . ]
+and the sequence
+�
+pn
+qn
+�
+n∈N0 =
+� 1
+1, 2
+1, 3
+2, 8
+5, 19
+12, 65
+41, 84
+53, 485
+306, . . .
+�
+. For r = 1+
+√
+5
+2
+, we have
+[1; 1, 1, 1, . . . ] and
+�
+pn
+qn
+�
+n∈N0 =
+� 1
+1, 2
+1, 3
+2, 5
+3, 8
+5, 13
+8 , 21
+13, 34
+21, . . .
+�
+.
+We will call the fractions pn
+qn the convergents of r. A semi-convergent of r is a
+number pn−1+ipn
+qn−1+iqn where i ∈ {0, 1, . . . , an+1}. Note that each convergent of r is also
+a semi-convergent of r. The names are motivated by the use of convergents and
+semi-convergents as rational approximations of an irrational number r.
+A rational number p
+q is a best approximation of an irrational number r, if any
+fraction p′
+q′ ̸= p
+q with q′ < q satisfies
+����q′
+�
+r − p′
+q′
+����� >
+����q
+�
+r − p
+q
+����� .
+A rational number p
+q is a best lower approximation of r if
+q′
+�
+r − p′
+q′
+�
+> q
+�
+r − p
+q
+�
+≥ 0
+for all rational numbers p′
+q′ with p′
+q′ ≤ r, p
+q ̸= p′
+q′, and 0 < q′ ≤ q. Similarly, p
+q is a
+best upper approximation of r if
+q′
+�
+r − p′
+q′
+�
+< q
+�
+r − p
+q
+�
+≤ 0
+18
+
+for all rational numbers p′
+q′ with p′
+q′ ≥ r, p
+q ̸= p′
+q′ , and 0 < q′ ≤ q.
+The first part of the following lemma is a classical result, the second and third
+parts are recent results of Hanˇcl and Turek [8].
+Lemma 16 ([8, 12]). Let r be a real number with r = [a0; a1, a2, . . . ] and let pn
+qn be
+the nth convergent of r for each n ∈ N0. Then, the following three statements hold.
+1. The set of best approximations of r consists of convergents pn
+qn of r.
+2. The set of best lower approximations of r consists of semi-convergents pn−1+ipn
+qn−1+iqn
+of r with n odd and 0 ≤ i < an+1.
+3. The set of best upper approximations of r consists of semi-convergents pn−1+ipn
+qn−1+iqn
+of r with n even and 0 ≤ i < an+1, except for the pair (n, i) = (0, 0).
+Finally, a real number r is restricted if there is a positive integer M such that
+all the partial denominators ai from the continued fraction of r are at most M.
+The restricted numbers are exactly those numbers r that are badly approximable
+by rationals, that is, there is a constant c > 0 such that for every p
+q ∈ Q we have
+���r − p
+q
+��� >
+c
+q2.
+We divide the rest of the proof of Theorem 6 into two cases, depending on
+whether logα(β) is restricted or not.
+6.3.2
+Unrestricted case
+First, we assume that logα(β) is not restricted.
+Let [a0; a1, a2, a3, . . . ] be the
+continued fraction of logα(β) with pn
+qn = [a0; a1, . . . , an] for every n ∈ N0. Then, for
+every positive integer m, there is a positive integer n(m) such that an(m)+1 ≥ m.
+We use this assumption to construct, for every positive integer m, a convex polygon
+with at least m vertices from L(α, β) that is empty in L(α, β).
+For a given m, consider the integer n(m) and let W be the set of points
+wi = (αpn(m)−1+ipn(m), βqn(m)−1+iqn(m))
+where i ∈ {0, 1, . . . , an(m)+1}. That is, we consider points where the exponents form
+semi-convergents
+pn(m)−1+ipn(m)
+qn(m)−1+iqn(m) to logα(β). We abbreviate pn,i = pn(m)−1 + ipn(m)
+and qn,i = qn(m)−1 + iqn(m). Observe that |W| ≥ m. We will show that W is the
+vertex set of an empty convex polygon in L(α, β). To do so, we assume without
+loss of generality that n(m) is even so that βqn(m)
+αpn(m) > 1. The other case when n(m)
+is odd is analogous.
+19
+
+First, we show that W is in convex position. In fact, we prove that all triples
+(wi1, wi2, wi3) with i1 < i2 < i3 are oriented counterclockwise. It suffices to show
+this for every triple (wi, wi+1, wi+2). To do so, we need to prove the inequality
+y(wi+2) − y(wi+1)
+x(wi+2) − x(wi+1) = βqn,i+2 − βqn,i+1
+αpn,i+2 − αpn,i+1 > βqn,i+1 − βqn,i
+αpn,i+1 − αpn,i = y(wi+1) − y(wi)
+x(wi+1) − x(wi).
+After dividing by βqn(m)−1
+αpn(m)−1 , this can be written as
+β(i+2)qn(m) − β(i+1)qn(m)
+α(i+2)pn(m) − α(i+1)pn(m) > β(i+1)qn(m) − βiqn(m)
+α(i+1)pn(m) − αipn(m) .
+If divide both sides by β(i+1)qn(m)−βiqn(m)
+α(i+1)pn(m)−αipn(m) , then the above inequality becomes
+βqn(m)
+αpn(m) > 1.
+This is true as n(m) is even.
+It remains to prove that the polygon Q with the vertex set W is empty in
+L(α, β). Suppose for contradiction that there is a point (αp, βq) of L(α, β) lying
+in the interior of Q. Let i be the minimum positive integer from {1, . . . , an(m)+1}
+such that q < qn,i. Such an i exists as (αp, βq) is in the interior of Q. We then have
+qn,i−1 < q < qn,i. Since (αp, βq) is in the interior of Q and W lies below the line
+x = y, we have p
+q > logα(β). So it is enough to prove that (αp, βq) does not lie
+above the line wi−1wi.
+We have pn,i −logα(β)qn,i < pn,i−1 −logα(β)qn,i−1 as pn,i
+qn,i is a best upper approx-
+imation of logα(β) and qn,i−1 < qn,i. This implies βqn,i−1
+αpn,i−1 < βqn,i
+αpn,i , or equivalently
+that wi lies above the line determined by wi−1 and the origin.
+Now if (αp, βq) lies above the line wi−1wi, then it also lies above the line
+determined by wi−1 and the origin. Thus, βqn,i−1
+αpn,i−1 < βq
+αp, implying
+p − logα(β)q < pn,i−1 − logα(β)qn,i−1,
+which means that p
+q is a better upper approximation of logα(β) than pn,i−1
+qn,i−1 . Thus,
+there exists a best upper approximation p∗
+q∗ of logα(β) with qn,i−1 < q∗ < qn,i. This
+contradicts part (c) of Lemma 16 as p∗
+q∗ is not a semi-convergent of logα(β).
+6.3.3
+Restricted case
+Now, assume that the number logα(β) is restricted. Let [a0; a1, a2, a3, . . . ] be the
+continued fraction of logα(β) with
+pn
+qn = [a0; a1, . . . , an] for every n ∈ N0. Let
+M = M(α, β) be a number satisfying
+an ≤ M
+(2)
+20
+
+for every n ∈ N0 and let c = c(α, β) > 0 be a constant such that
+����logα(β) − p
+q
+���� > c
+q2
+(3)
+holds for every p
+q ∈ Q. Recall that αpn
+βqn < 1 for even n and αpn
+βqn > 1 for odd n. Note
+also that the sequence
+�
+αpn
+βqn
+�
+n∈N0 converges to 1 as
+�
+pn
+qn
+�
+n∈N0 converges to logα(β).
+Moreover, the terms of
+�
+pn
+qn
+�
+n∈N0 with odd indices form a decreasing subsequence
+and the terms with even indices determine an increasing subsequence.
+Let n0 = n0(α, β) be a sufficiently large positive integer and let V be the set of
+points vn = (αpn, βqn) for every odd n ≥ n0. Note that V is a subset of L(α, β).
+We first show that V is in convex position. In fact, we prove a stronger claim
+by showing that the orientation of every triple (vn1, vn2, vn3) with n1 < n2 < n3 is
+counterclockwise. It suffices to show this for every triple (vn−4, vn−2, vn). To do so,
+we prove that the slopes of the lines determined by consecutive points of V are
+increasing, that is,
+y(vn) − y(vn−2)
+x(vn) − x(vn−2) = βqn − βqn−2
+αpn − αpn−2 > βqn−2 − βqn−4
+αpn−2 − αpn−4 = y(vn−2) − y(vn−4)
+x(vn−2) − x(vn−4)
+for every even n ≥ n0. By dividing both sides of the inequality with βqn−2
+αpn−2 , we
+rewrite this expression as
+βqn−qn−2 − 1
+αpn−pn−2 − 1 > 1 − βqn−4−qn−2
+1 − αpn−4−pn−2 .
+Using (1), this is the same as
+βanqn−1 − 1
+αanpn−1 − 1 > 1 − β−an−2qn−3
+1 − α−an−2pn−3 .
+The above inequality can be rewritten as
+(βanqn−1 − 1)(1 − α−an−2pn−3) > (αanpn−1 − 1)(1 − β−an−2qn−3),
+where βqn−1 > αpn−1 > 1 and 1 > α−pn−3 > β−qn−3 > 0 as n − 1 and n − 3 are even.
+Therefore, if the above inequality holds for an = 1 = an−2, then it holds for any an
+and an−1 as both numbers are always at least 1. Thus, it suffices to show
+(βqn−1 − 1)(1 − α−pn−3) > (αpn−1 − 1)(1 − β−qn−3).
+(4)
+We prove this using the following simple auxiliary lemma.
+21
+
+Lemma 17. Consider the function f : R+ × R+ → R given by f(x, y) = (x −
+1)(1 − 1/y). Let x, y, x′, y′ > 1 be real numbers such that 1 − 1
+y − x
+x′ > 0. Then,
+f(x′, y) > f(x, y′).
+Proof. We have
+f(x′, y) − f(x, y′) = (x′ − 1)
+�
+1 − 1
+y
+�
+− (x − 1)
+�
+1 − 1
+y′
+�
+= x′ − x′ − 1
+y
+− x + x − 1
+y′
+> x′ − x′
+y − x = x′
+�
+1 − 1
+y − x
+x′
+�
+> 0,
+where the last inequality follows from 1 − 1
+y − x
+x′ > 0.
+Now, by choosing x = αpn−1, x′ = βqn−1, y = αpn−3, and y′ = βqn−3, the
+inequality (4) becomes f(x′, y) > f(x, y′). In order to prove it, we just need to
+verify the assumptions of Lemma 17. We clearly have x, x′, y, y′ > 1. It now suffices
+to show 1 − 1
+y − x
+x′ > 0. By (3), we obtain that qn−1 logα(β) − pn−1 ≥ c/qn−1, thus
+x
+x′ = αpn−1
+βqn−1 ≤ α−c/qn−1.
+Now, to bound qn−1 in terms of pn−3, equation (1) gives
+qn−1 = an−1qn−2 + qn−3 ≤ (M + 1)qn−2 = (M + 1)(an−2qn−3 + qn−4)
+≤ (M + 1)2qn−3 ≤ 2 logβ(α)(M + 1)2pn−3,
+where we used (2) and qn−4 ≤ qn−3 ≤ qn−2, qn−3 ≤ 2 logβ(α)pn−3 for n large enough.
+It follows that qn−1 ≤ M ′pn−3 for a suitable constant M ′ = M ′(α, β) > 0. Thus,
+1 − 1
+y − x
+x′ ≥ 1 − α−pn−3 − α−c/qn−1 ≥ 1 − α−pn−3 − α−c/(M′pn−3),
+which is at least
+c ln α
+2M ′pn−3
+−
+1
+αpn−3
+as 1−c ln α/(2M ′pn−3) ≥ e−2c ln α/(2M′pn−3) = α−c/(M′pn−3) if 0 < c ln α/(2M ′pn−3) <
+1/2. The last expression is positive if n ≥ n0 and n0 is sufficiently so that pn−3 is
+large enough.
+It remains to show that the convex polygon P with the vertex set V is empty
+in L(α, β). We proceed analogously as in the unrestricted case. Suppose for
+contradiction that there is a point (αp, βq) of L(α, β) lying in the interior of P.
+Then, let vn = (αpn, βqn) be the lowest vertex of P that has (αp, βq) below. Such a
+vertex vn exists, as V contains points with arbitrarily large y-coordinate. By the
+22
+
+choice of vn, we obtain qn−2 < q < qn. Since (αp, βq) is in the interior of P and V
+lies below the line x = y, we have p
+q > logα(β) > pn−1
+qn−1 . Moreover, since all triples
+from V are oriented counterclockwise, the point (αp, βq) lies above the line vn−2vn.
+Let
+wi = (αpn−2+ipn−1, βqn−2+iqn−1)
+where i ∈ {0, 1, . . . , an−1} similarly as in the proof of the unrestricted case. There,
+it was shown that all the triples wi−1, wi, wi+1 are oriented counterclockwise, thus
+all the points wi with i ∈ {1, . . . , an−1 − 1} lie below the line vn−2vn. Thus, if
+(αp, βq) lies above the segment connecting vn−2 and vn, then there is an i such
+that (αp, βq) lies above the segment connecting wi−1 and wi. As in the last two
+paragraphs of the proof of the unrestricted case, the position of (αp, βq) implies
+the inequality p − logα(β)q < pn,i−1 − logα(β)qn,i−1, and the contradiction follows
+from part (c) of Lemma 16, as there can be no best upper approximation of logα(β)
+which is not a semi-convergent of logα(β).
+Acknowledgment
+This research was initiated at the 11th Eml´ekt´abla workshop
+on combinatorics and geometry. We would like to thank G´eza T´oth for interesting
+discussions about the problem during the early stages of the research.
+References
+[1] Nina Amenta, Jes´us A. De Loera, and Pablo Sober´on. Helly’s theorem: new
+variations and applications. In Algebraic and geometric methods in discrete
+mathematics, volume 685 of Contemp. Math., pages 55–95. Amer. Math. Soc.,
+Providence, RI, 2017.
+[2] David E. Bell. A theorem concerning the integer lattice. Studies in Appl.
+Math., 56(2):187–188, 1976/77.
+[3] Jes´us A. De Loera, Reuben N. La Haye, D´eborah Oliveros, and Edgardo
+Rold´an-Pensado. Helly numbers of algebraic subsets of Rd and an extension
+of Doignon’s theorem. Adv. Geom., 17(4):473–482, 2017.
+[4] Jes´us A. De Loera, Reuben N. La Haye, David Rolnick, and Pablo Sober´on.
+Quantitative Tverberg theorems over lattices and other discrete sets. Discrete
+Comput. Geom., 58(2):435–448, 2017.
+[5] Travis Dillon. Discrete quantitative Helly-type theorems with boxes. Adv. in
+Appl. Math., 129:Paper No. 102217, 17, 2021.
+23
+
+[6] Jean-Paul Doignon. Convexity in cristallographical lattices. J. Geom., 3:71–85,
+1973.
+[7] Alexey Garber. On Helly number for crystals and cut-and-project sets. Arxiv
+preprint arxiv.org/abs/1605.07881, 2017.
+[8] Jaroslav Hanˇcl and Ondˇrej Turek. One-sided Diophantine approximations.
+Journal of Physics A: Mathematical and Theoretical, 52(4):045205, jan 2019.
+[9] Eduard Helly. ¨Uber Mengen konvexer K¨orper mit gemeinschaftlichen Punkten.
+Jahresber. Deutsch. Math.-Verein., 32:175–176, 1923.
+[10] Alan J. Hoffman. Binding constraints and Helly numbers. In Second Interna-
+tional Conference on Combinatorial Mathematics (New York, 1978), volume
+319 of Ann. New York Acad. Sci., pages 284–288. New York Acad. Sci., New
+York, 1979.
+[11] Andreas Holmsen and Rephael Wenger. Helly-type theorems and geometric
+transversals. In Handbook of Discrete and Computational Geometry (3rd ed.).
+CRC Press, 2017.
+[12] Aleksandr Ya. Khinchin.
+Continued fractions.
+Dover Publications, Inc.,
+Mineola, NY, Russian edition, 1997. With a preface by B. V. Gnedenko,
+reprint of the 1964 translation.
+[13] Herbert E. Scarf. An observation on the structure of production sets with
+indivisibilities. Proc. Nat. Acad. Sci. U.S.A., 74(9):3637–3641, 1977.
+[14] Kevin Barrett Summers. The Helly Number of the Prime-coordinate Point
+Set. Bachelor’s thesis, University of California, 2015.
+24
+

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+Controlling unpredictability in the randomly driven
+Hénon–Heiles system
+Mattia Coccolo ⇑, Jesús M. Seoane, Miguel A.F. Sanjuán
+Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
+a r t i c l e
+i n f o
+Article history:
+Received 20 November 2012
+Received in revised form 8 May 2013
+Accepted 9 May 2013
+Available online 22 May 2013
+Keywords:
+Hénon–Heiles
+Unpredictability
+Wada basins
+Control
+Non-linear dynamics
+Chaotic scattering
+a b s t r a c t
+Noisy scattering dynamics in the randomly driven Hénon–Heiles system is investigated in
+the range of initial energies where the motion is unbounded. In this paper we study, with
+the help of the exit basins and the escape time distributions, how an external perturbation,
+be it dissipation or periodic forcing with a random phase, can enhance or mitigate the
+unpredictability of a system that exhibit chaotic scattering. In fact, if basin boundaries have
+the Wada property, predictability becomes very complicated, since the basin boundaries
+start to intermingle, what means that there are points of different basins close to each
+other. The main responsible of this unpredictability is the external forcing with random
+phase, while the dissipation can recompose the basin boundaries and turn the system more
+predictable. Therefore, we do the necessary simulations to find out the values of dissipation
+and external forcing for which the exit basins present the Wada property. Through these
+numerical simulations, we show that the presence of the Wada basins have a specific rela-
+tion with the damping, the forcing amplitude and the energy value. Our approach consists
+on investigating the dynamics of the system in order to gain knowledge able to control the
+unpredictability due to the Wada basins.
+� 2013 Elsevier B.V. All rights reserved.
+1. Introduction
+There exist a lot of theoretical and experimental works, investigating responses of dynamical systems to external pertur-
+bations, such as noise, dissipation or periodic forcing. Depending on the properties of the dynamical systems and the applied
+perturbation, responses can vary extremely, ranging from practically no effects to a suppressed or an enhanced response [1],
+regularization of chaotic states [2], chaotification [3], or control of chaotic dynamics [4], among others.
+One of the physical phenomenon that exhibits this kind of behavior is the chaotic scattering phenomenon. Chaotic scat-
+tering is usually associated with the Hamiltonian equations of motion, that are actually related with chaotic processes. Nor-
+mally, in this kind of systems, there exists a threshold value of the energy, the escape energy, beyond which the trajectories
+are unbounded and several exits may appear, therefore the particles are able to leave the scattering region. Since a trajectory
+might leave the potential well, these systems are usually called open Hamiltonian systems. In these cases the particle bounces
+back and forth in a bounded region, the scattering region, for a certain time before eventually escaping the region towards
+the infinity.
+The phenomenon of chaotic scattering in open Hamiltonian systems has been studied for several years since it has a lot of
+applications in different fields in science and engineering [5]. Some applications are the analysis of escape from galaxies [6],
+the study of the interaction between the Earth and the solar wind [7] and many others.
+1007-5704/$ - see front matter � 2013 Elsevier B.V. All rights reserved.
+http://dx.doi.org/10.1016/j.cnsns.2013.05.009
+⇑ Corresponding author.
+E-mail address: [email protected] (M. Coccolo).
+Commun Nonlinear Sci Numer Simulat 18 (2013) 3449–3457
+Contents lists available at SciVerse ScienceDirect
+Commun Nonlinear Sci Numer Simulat
+journal homepage: www.elsevier.com/locate/cnsns
+
+ELSEVLERCommunicationsin
+Nonlinear
+Science and
+NumericalSimulationOn the other hand, in the case of a conservative Hamiltonian system, the total energy is preserved, and thus, it is not pos-
+sible to talk about attractors nor basins of attraction. A basin of attraction is defined as the set of points that, taken as initial
+conditions, are attracted to a specific attractor [8]. When we can define two different attractors in a certain region of phase
+space, two basins exist, which are separated by a basin boundary. This basin boundary can be a smooth curve or can be in-
+stead a fractal curve. While we cannot talk about attractors in Hamiltonian systems, we can however define exit basins in an
+analogous way to the basins of attraction in a dissipative system. In our case, an exit basin is the set of initial conditions that
+lead to a certain exit.
+When boundaries are complicated in a specific region of initial points, a small uncertainty in the position of the initial
+conditions may yield a greater uncertainty in order to detect the final exit of the trajectory. In fact, there are situations where
+a small uncertainty in the initial conditions can make them to belong to any of the basins. Nothing can be said because any
+point on the boundary is arbitrary close to points in all the basins. In the case where we have multiple destinations for the
+scattering trajectories, the structure of the basins can, eventually, be more complicated [9] and might show the Wada prop-
+erty [10]. A basin B verifies the property of Wada if any boundary point also belongs to the boundary of two other basins. In
+other words, every open neighborhood of a point x belonging to a Wada basin boundary has a nonempty intersection with at
+least three different basins [11]. Hence, if the initial conditions of a particle are in the vicinity of a Wada basin boundary, we
+will not be able to be sure by which one of the three exits the trajectory will escape to infinity.
+It has been proved that the Wada property can be found in a triangular configuration [11], and typically appears in chaotic
+scattering systems. Some experimental evidence has been reported in Refs. [12,13] where Wada basins are apparent for
+higher dimensions. Then, as we said before, the external perturbations can enhance deep modifications in the structures
+of the basins of the system. In fact, in an open Hamiltonian system, where chaotic scattering phenomena are important,
+the effects of the dissipation have been an interesting topic [14] because they can induce a new kind of dynamics in the sys-
+tems [15–18]. Thus, if we add an external perturbation, the topology of the phase space can change abruptly, with the pres-
+ence of new basins also with the Wada property, as we can see in Ref. [19].
+By way of explanations, it is interesting to investigate in which form the influence of an externally driven perturbation
+and a dissipation term can change the dynamics of a chaotic system. From what we have written above, an external forcing
+and a damping, can make the system more or less complicated. In other words, the possibility to directly operate on the
+intensity of an external forcing as a function of the dissipation can reduce the roughness of the basin boundaries until
+the disappearance of some unpredictabilities, associated to fractal or Wada basins. In the current work, we focus our interest
+in the numerical analysis of the damped and forced Hénon–Heiles Hamiltonian [20], which is a model of an axisymmetrical
+galaxy that exhibit chaotic scattering. This is a two dimensional time-independent dynamical system, that shows three dif-
+ferent exits for energies greater than the escape energy, so that the system possesses the chaotic scattering phenomenon. In
+this system we have implemented dissipation and a noisy driven external excitation, in order to study their influence on the
+topology of the system. To summarize, our goal in this paper is to study the dependence of the Wada basins on the damping
+and the forcing with a random phase which include the presence of noise [23,22,21]. In other words, we study the possibility
+to control these unpredictabilities of the system by applying weak external perturbations.
+The organization of the paper is as follows. In Section 2 we study the model and the nature of the trajectories. In Sec 3.1
+we investigate the external perturbation influence on the unpredictability of the system. In Section 3.2 we investigate the
+Wada property of the exit basins, changing the bounded excitation and the dissipation at a constant energy and we analyze
+our data. Finally, a discussion and the main conclusions of this paper are summarized in Section 4.
+2. Model description
+In order to show the influence of an external perturbation on a system with chaotic scattering we take as a prototype
+model, the Hénon–Heiles Hamiltonian [20], written as
+H ¼ 1
+2 ð_x2 þ _y2Þ þ 1
+2 ðx2 þ y2Þ þ x2y � 1
+3 y3:
+ð1Þ
+For energies below the escape energy Ee ¼ 1=6, trajectories are bounded and consequently there are no exits. For the energy
+Ee ¼ 1=6, the equipotential line is an equilateral triangle, which is the limit energy at which the motion is bounded as shown
+in Fig. 1(a). On the other hand, if the energy is larger than this threshold value, the system has three exits with a 2p=3 rota-
+tion symmetry, from which the trajectories may escape and go to infinity as shown in Fig. 1(b). Due to the symmetry of the
+system, the three exits are: exit 1 ðy ! þ1Þ, exit 2 ðy ! �1; x ! �1Þ and exit 3 ðy ! �1; x ! þ1Þ, which are plotted in
+Fig. 1(b). In this case, there exist three orbits Liði ¼ 1; 2; 3Þ, known as Lyapunov orbits, one corresponding to each exit, acting
+as frontiers: any trajectory that crosses them with an outward-oriented velocity must go to infinity and never come back. We
+focus our study in a situation with escapes from the scattering region, so from now on we use values of E > Ee. We study the
+Hénon–Heiles system subjected to a bounded noisy excitation (a periodic forcing with a random phase) [19] and a dissipa-
+tion proportional to the velocity [17]. The equations of motion can be written as
+€x þ x þ 2xy þ a_x ¼ 0
+ð2Þ
+€y þ y þ x2 � y2 þ b_y ¼ f cos½Xt þ rBðtÞ þ c�;
+ð3Þ
+3450
+M. Coccolo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 3449–3457
+
+where a and b are damping coefficients, f and X are the amplitude and frequency of the external excitation, respectively, BðtÞ
+is the standard Wiener process with the amplitude r, and c is a random variable uniformly distributed in the interval ½0; 2pÞ.
+When a ¼ b ¼ f ¼ 0 we can recognize the Hénon–Heiles conservative system. From now on, and without any loss of gener-
+ality, we take a ¼ b ¼ l as dissipative parameter.
+We are studying a two-dimensional time-independent Hamiltonian, so the phase space depends on ðx; y; _x; _yÞ and one
+conserved quantity, the energy E. Throughout this paper, we will use a Poincaré surface of section to show our results.
+For that purpose, our choice is ðx ¼ 0; y; _yÞ. Thus, the dynamical description of the system can be reduced to a study of
+the ðy; _yÞ surface. Naturally, the equation of the initial velocity, generically expressed by
+vi ¼
+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
+_x2 þ _y2
+p
+¼
+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
+2E � x2
+i � y2
+i � 2x2
+i yi þ 2=3y3
+i
+q
+ð4Þ
+becomes vi ¼
+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
+2E � y2
+i � 2=3y3
+i
+q
+. This simplification is only valid for the initial time t ¼ 0, when the dissipation and the
+external forcing are not yet acting.
+In order to study the phase space structure for the Hénon–Heiles Hamiltonian, we compute the exit basins. For that pur-
+pose, we compute each trajectory for a large number of initial conditions, ðy; hÞ, where h, the shooting angle, is the initial angle
+between the y axis and the trajectory, as shown in Fig. 2. In this way, we can start the trajectories on all the points of the
+Poincaré section, x ¼ 0, and calculate the exit through which every trajectory leaves the potential well. Therefore, knowing
+the initial conditions related to every trajectory, we color them in a different way, according to the exit through which the
+trajectories leave the potential, as shown in Fig. 3(a) and (b). We calculate the trajectories and compute the exit basins by
+using a symplectic integrator (SI), that can be mathematically defined as a numerical integration scheme for a specific group
+of differential equations related to classical mechanics and symplectic geometry. Symplectic integrators form the subclass of
+geometric integrators that, by definition, are canonical transformations. They are widely used in molecular dynamics, finite
+element methods, accelerator physics, and celestial mechanics. The trajectories of each particle is followed by numerically
+solving the Hamiltonian equations from the C-C Algorithm, which is a fourth-order forward symplectic algorithm proposed
+recently by Chin and Chen [24]. This algorithm can follow the true dynamics longer because it can preserve the symplectic
+structures of the Hamiltonian equations.
+Fig. 1. (a) This figure represents the isopotential curves of the Hénon–Heiles system for different values of the energy, in which both bounded and
+unbounded motions can take place. (b) Plot of the isopotential curve for the unbounded case for energy value E ¼ 0:21.
+Fig. 2. This figure represents the shooting angle h of a typical trajectory inside the Hénon–Heiles potential.
+M. Coccolo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 3449–3457
+3451
+
+(a)
+(b)
+Exit 1
+0.5
+0.5
+0
+0
+-0.5
+Exit 2
+Exit 3
+0.5
+-1
+-0.5
+0
+0.5
+-1
+-0.5
+0
+0.5
+X
+X1
+0.5
+0
+0
+-0.5
+-11
+-0.5
+0
+0.5
+XAs we said earlier in this section, we integrate within the variables of the position, q, and the momentum, p, as we can see
+in Fig. 4(a) and (b), including the noisy part of the external excitation,
+rBðtÞ ¼
+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
+�4D logðr1Þ sinð2pr2Þ
+q
+;
+ð5Þ
+where r ¼
+ffiffiffiffiffiffiffi
+4D
+p
+and BðtÞ ¼
+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
+� logðr1Þ sinð2pr2Þ
+p
+of Eq. (3), r1 and r2 are random numbers in the interval ð0; 1Þ. It is possible to
+appreciate in Fig. 4(a), a trajectory affected only by dissipation, while in Fig. 4(b) the trajectory is affected also by the noisy
+excitation. Both trajectories start in the same initial condition and are affected by the same amount of dissipation, the only
+change we introduced is the amount of noise in the random phase of the external forcing.
+3. Numerical results
+In this Section, we are going to provide numerical evidence on the effects of the external perturbation in both the dynam-
+ics and the topology of the system and how we can tame the effects of both perturbations in order to reduce the unpredict-
+ability of the randomly driven Hénon–Heiles system.
+Fig. 3. (a) The figure represents the exit basins for dissipative parameter value l ¼ 0:06, without forcing. (b) The exit basins with both damping, l ¼ 0:06,
+and forcing, f ¼ 0:008 are plotted. Both figures show the exit basins and each color denotes the exit through which trajectories with that initial condition
+escape: exit 1 (blue, ðy ! þ1Þ), exit two (red, ðy ! �1; x ! �1Þ) and exit 3, (yellow ðy ! �1; x ! þ1Þ). White color inside the color structure denotes
+the points that do not leave from the scattering region. (For interpretation of the references to color in this figure legend, the reader is referred to the web
+version of this article.)
+Fig. 4. (a) This figure represents a trajectory with dissipation, l ¼ 0:07 and energy E ¼ 0:25 without forcing, with initial conditions ðx0; y0Þ ¼ ð0; 0Þ and
+shooting angle, so called the initial angle between the y axis and the trajectory, h ¼ 0:45p. (b) A trajectory with both damping, l ¼ 0:07, and forcing
+amplitude f ¼ 0:045, and the same initial conditions as in Fig. 4(a) is plotted. We easily observe the effects of the noisy excitation since this trajectory is
+similar to a random walk escaping the particle through exit three after a long time.
+3452
+M. Coccolo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 3449–3457
+
+1.5
+1.5
+(a)
+(b)
+1
+0.5
+0.5
+0
+0
+-0.5
+-0.5
+-1
+-1
+-1.5
+-1.5
+-1
+0
+1
+2
+-1
+0
+1
+2
+Y1.2
+1.2
+(a)
+(b)
+0.8
+0.8
+0.6 
+0.6 
+0.4
+0.4
+0.2
+0.2
+0
+0
+-0.2
+-0.2
+-0.4
+-0.4
+-0.6
+-0.6
+0.8
+-0.8
+-0.5
+-0.5
+0
+0.5
+-1
+0
+0.5
+1
+X
+X3.1. The effect of the external perturbations on the dynamics of the system
+One of the main consequences of chaotic scattering in the Hénon–Heiles system is that, a trajectory may spend a long-
+time wandering in the vicinity of the scattering region before escaping to infinity from one of the three exits. The transient
+chaotic dynamics inside the scattering region is governed by the nonattracting chaotic set, also known as the chaotic saddle.
+This set can be computed through the intersection of the stable manifold that contains the trajectories that will never escape
+for t ! þ1, and the unstable manifold that contains the ones that will never escape for t ! �1. Both of these sets have
+singularities, therefore their dimension is fractal [18]. Due to the sensitivity to the initial conditions, characteristic of the cha-
+otic systems, particles can exhibit dramatically different asymptotic behavior. Moreover, if we include dissipation and an
+external forcing to the system, the exit basins can also change drastically. On the other hand, the phase space might be mixed
+with KAM islands and chaotic seas, and including a small amount of dissipation can convert the elliptic points inside the
+islands into sinks, or attractors [15,14]. These dissipation-induced basins of attraction can be intermingled in complicated
+ways as well, leading to unpredictability or a well defined final state, depending on the initial condition and on the external
+forcing applied. If we vary the values of the damping and the forcing, the exit basins and the dissipation-induced basins of
+attractions can show different levels of unpredictability. Basically, it can become difficult to define the exit by which a par-
+ticle would leave the scattering region, given a set of initial conditions. When the dissipation is high enough and the forcing
+is low enough, the basin boundaries are smooth. Topologically, this means that the basins are connected and compact. On the
+other hand, when the external excitation grows up, the basin boundaries become rough and the basins start to mix until they
+loose the connectedness and compactness. When this happens the boundaries start showing the Wada property and as a
+consequence the unpredictability in the evolution of the system increases. On the other hand, we focus our research on
+the unpredictability in a scattering problem in presence of a noisy excitation and dissipation. Here, we investigate the rela-
+tion between damping, forcing and their effects on the unpredictability in the Hénon–Heiles system. This study is carried out
+for different values of the energy always beyond the critical energy Ee ¼ 0:16 which separates bounded and unbounded tra-
+jectories. Therefore, we want to analyze the control of the unpredictability due to forcing and noise, through the energy dis-
+sipation. This means that we need to show where the Wada basins appear in function of the energy, the dissipation and the
+forcing. Naturally, in order to study the above relations, it is important to understand the role of the random phase of the
+Fig. 5. Figures (a) and (c) show the intensity of the noise, as shown in Eq. (5), rBðtÞ ¼
+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
+�4D logðr1Þ sinð2pr2Þ
+p
+, with respectively D ¼ 10�6 and D ¼ 10�2,
+while r1; r2 are random numbers chosen in the interval ½0; 1�. Figures (b) and (d) show the intensity of forcing, as shown in the right hand of Eq. (3),
+f cos½Xt þ rBðtÞ þ c�, with respectively D ¼ ðr=2Þ2 ¼ 10�6 and D ¼ 10�2, f ¼ 0:04; X ¼ 1 and c is a random number chosen in the interval ½0; 2pÞ.
+M. Coccolo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 3449–3457
+3453
+
+B(t)
+Forcing
+0.04
+0.35
+(a)
+(b)
+0.03
+0.3
+intensity of forcing
+intensity of B(t)
+0.02
+0.25
+0.01
+0.2
+0.15
+0.0
+0. 
+-0.02
+0.03
+0.05
+-0.04,
+0
+2000
+4000
+6000
+8000
+10000
+2000
+4000
+6000
+8000
+10000
+number of iterations
+number of iterations
+B(t)
+Forcing
+5
+(c)
+(d)
+0.3
+3
+intensity of forcing
+25
+intensity of B(t)
+0.2 
+0.15
+0.1
+3
+0
+0
+2000
+4000
+6000
+8000
+10000
+0
+2000
+4000
+6000
+8000
+number of iterations
+numberofiterationsexternal excitation. We have decided to perturb our system, Eq. (2), with a bounded noisy excitation, that is, a periodic force
+with a random phase. The value D, in Eq. (5), slightly affects the trajectory with a small amplitude oscillation that depends on
+the amplitude of the forcing f, and the fluctuation of the function cos½Xt þ rBðtÞ þ c�, as we can see in Fig. 5(a)–(d). In these
+figures, in fact, we show how both the forcing and the noisy excitation act on the system. Fig. 5(a) and (c) show the intensity
+of the noise every iteration. The difference between the figures is the amplitude of the noise D and its effects on the intensity
+of the signal. Actually, it is possible to appreciate in the graphs the difference of magnitude in the scales. The other two fig-
+ures (b) and (d) show the intensity of the external forcing, with the same amplitude f, but with the two previous noise signals
+inside. In other words, those figures show a typical effect of the bounded noise. Its use assures us that, even if integrated
+along with ðx; yÞ, it never overcomes the trajectories of the system but only affects them as a bounded perturbation.
+3.2. Computing the appearance of the Wada property in the exit basin in function of the external excitation and the dissipation
+As we discussed earlier, the capacity of predicting the behavior of a system is crucial in science and engineering, so when
+some unpredictabilities show up in the system, their control becomes important. In this section, we analyze by numerical
+simulations, how the damping can help us to recompose the basins and reduce the merging of the basins. To achieve this
+goal, we calculate the basins keeping constant the initial energy, E ¼ 0:25, and changing both the dissipative parameter
+and the noisy excitation, evaluating every single case. When the external forcing and the damping are changed, we have seen
+that it is possible to discern when the structure of the basins looses the coherence and the boundaries start to mix. For bigger
+values of dissipation and a lower value of the forcing, the basin structures are connected and compact, as shown in Fig. 6(a).
+While when we decrease the damping and increase the forcing the same boundaries start to intermingle as we see in
+Fig. 6(b). Now, in order to distinguish the cases between Wada and non-Wada a formal method is needed and it is provided
+by the theorem of Kennedy and Yorke [25]. It states that, if P is a periodic point on the basin boundary, the following two
+Fig. 6. (a) The figure represents respectively, for E ¼ 0:25, the basins of the system for an amount of forcing f ¼ 0:0005 and damping l ¼ 0:05, for which the
+boundaries are coherent. (b) The figure plots respectively, E ¼ 0:25, the basins of the system for an amount of forcing f ¼ 0:005 and damping l ¼ 0:05, for
+which the boundaries are mixed. Here the influence of a bigger external forcing can be observed, making the basins more intermingled than in Fig. 6(a). (c)
+The figure shows the unstable manifold, the black curve, of the Lyapunov orbit drawn on a zoom of the basins of Fig. 6(b). It is possible to see that the
+unstable manifold intersects all the basins, so the Wada property is satisfied.
+3454
+M. Coccolo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 3449–3457
+
+1.5
+1.5
+μ=0.05,f=0.0005.NON-wADA
+μ=0.05,f=0.005,WADA
+1
+0.5
+0.5
+0
+0
+-0.5
+-0.5
+-1
+(a)
+(q)
+-1.5,
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+2
+-1
+-0.5
+0
+0.5
+1
+1.5
+2
+Y
+Y
+0.2
+lyapunov Orbit
+0
+-0.3
+-0.6
+(c)
+-0.6
+0
+1
+2conditions are satisfied: (1) its unstable manifold intersects every basin (main condition), and (2) this is the only periodic
+point accessible from the basin of interest, then the basins have the Wada property. This last property, for the Hénon–Heiles
+system, has been shown in Ref. [26]. Concerning to the main property, we show in Fig. 6(c) the unstable manifold of the peri-
+odic point P ¼ ð1:02461; 0Þ, representation of a Lyapunov orbit on the phase space ðy; _yÞ, and it intersects all the basins, ver-
+ifying the conditions of the Kennedy–Yorke theorem. On the other hand, Fig. 6(a) represents the basins of the system for an
+amount of dissipation and forcing from the non-Wada region.
+Nevertheless, there is a minimum value of the dissipation for which the basins are not intermingled. Below this value, the
+basins become Wada even without an external forcing. For this case, the unpredictability of the system increases and the
+prediction of its evolution becomes impossible. Starting from this point, we increased the dissipation and the excitation
+to find the limit in which Wada basins appear as shown in Fig. 7.
+Thus, two regions appear: the Wada region above the points and the non-Wada region below them. In the non-Wada re-
+gion, we can predict the evolution of the system while in the Wada region the basin topology is very complicated and the
+evolution of the system is quite difficult to figure out. The figure also shows on the top right a kind of plateau, as a conse-
+quence of a quasi-equilibrium of the external excitation with the damping.
+In Fig. 8(a) and (b) we plot the escape time for y ¼ 0, l ¼ 0:06 and different shooting angles. The difference between the
+figures is the intensity of the external forcing f, the first one belonging to the Wada region with a forcing value of
+f ¼ 5 � 10�3, while the second one to the non-Wada region, with f ¼ 5 � 10�4.
+It is possible to see the difference between the mean escape time, where Fig. 8(a) shows a smaller mean escape time than
+Fig. 8(b). Therefore, as we have thought, Wada basins are related with a smaller mean escape time.
+After having computed other escape times in the Wada and non-Wada regions, below and above the points shown in
+Fig. 7, we have found a similar trend. In fact, we obtain more or less the same results that we have shown in Fig. 8(a)
+and (b) as the forcing increases.
+Fig. 7. This figure represents, for an energy value E ¼ 0:25, the points of the damping-forcing plane for which the Wada property starts to appear in the exit
+boundaries. The points limit two regions: above them we have the region where Wada basins appear and below the region where we can not find Wada
+basins. In the non-Wada region, we can predict the evolution of the system while in the Wada region the basin topology is very complicated and the
+evolution of the system is quite difficult to figure out.
+Fig. 8. (a) and (b) Both figures represent the escape time for an E ¼ 0:25, with dissipation l ¼ 0:06, versus the shooting angle h=2p. In figure (a) forcing
+amplitude is f ¼ 5 � 10�3 and in figure (b) forcing amplitude is f ¼ 5 � 10�4. The amplitude of the noisy excitation helps the particles to escape from the
+scattering region as observed in panel (a). Note that the mean escape time (dash-dot line) is smaller in panel (a) than in panel (b).
+M. Coccolo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 3449–3457
+3455
+
+Wada
+11
+10
+1
+Forcing
+TI
+9
+8
+TI
+non-Wada
+7
+1
+1.
+0.04
+0.06
+0.08
+0.1
+Dampingμ=0.06,f=5×10-3
+μ=0.06,f=5*10-4
+250
+250
+(b)
+(a)
+200
+200
+Escape time
+150
+150
+100
+100
+mean escape time
+mean escape time
+50
+50
+00
+0.2
+0.4
+0.6
+0.8
+0.2
+0.4
+0.6
+0.8
+0/2元
+0/2元We have also studied the escape times varying the amplitude of the external excitation for a fixed amount of dissipation,
+l ¼ 0:06, and for a value of the energy E ¼ 0:25, as depicted in Fig. 9. Here, we can see that increasing the external forcing for
+values beyond 0:001, the mean escape time decreases strongly. As a consequence, the higher forcing amplitude implies the
+lower mean escape times since the particles escape faster from the scattering region insofar the forcing amplitude f in-
+creases. Actually, beyond that value the basins start to intermingle faster, so that the unpredictability increases, i.e., the dis-
+sipation-induced basins start to scatter. While the forcing helps the trajectories to leave the scattering region, the bounded
+noise produces a mixing in the basins and the mean escape times decrease.
+We have discussed earlier that it is possible to find a minimum value for the dissipation, for which the basins do not show
+the Wada property. So we have decided to investigate the relation between this minimum and the energy. We consider here
+a large range of initial energy values for which the motions are unbounded, between 0:19 and 0:3, and we use a forcing f ¼ 0
+in order to analyze the relationship between the damping and the initial energy.
+We show this relationship in Fig. 10, where we observe that as the initial energy increases, the minimum value of the
+damping also increases. This polynomial-like curve of the uncertainty boundary U, has a quadratic fit l � E2, and its math-
+ematical expression is given by l ¼ 1:3E2 � 0:21E þ 0:023. A possible explanation of this phenomenon lies in the integration
+equations in which the dissipation is a factor of the velocity, the equation of which is vi ¼
+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
+2E � y2
+i � 2=3y3
+i
+q
+. However, the
+effect of the dissipation on the system has to be the same for all the energies, that is to crash the connectedness and com-
+pactness of the basins in order to induce the Wada property even without the external excitation, f ¼ 0. Therefore, if in Eq.
+(2) the forcing is equal to zero, the damping value has to grow up with the energy as a factor of the velocity in a polynomial
+way, through Eq. (4) as depicted in Fig. 10.
+Even if, by looking at the tendency curve, it is possible to observe that it matches very well with the data, we report some
+statistical evidence of the goodness of this fit, like the correlation R2 ¼ 0:997 and the root mean square error rmse ¼ 0:00088.
+4. Conclusions
+We have studied in detail the dynamics of the randomly driven and dissipative Hénon–Heiles Hamiltonian. We consider
+the system subjected to dissipation and a random driven forcing, in the range of initial energy values higher than the escape
+Fig. 9. The figure shows the evolution of the mean escape time for a constant amount of dissipation, l ¼ 0:06, and energy, E ¼ 0:25, with respect to the
+variation of the external excitation. As it can be observed, the higher the forcing amplitude implies the lower mean escape time, since the particles escape
+faster from the scattering region insofar the forcing amplitude f increases.
+Fig. 10. The figure shows the relation between the minimum value of the damping and the initial energy for which Wada basins appear, where there is no
+external excitation, f ¼ 0. We can observe a quadratic fit, l � E2, which separates both regions, non-Wada and Wada.
+3456
+M. Coccolo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 3449–3457
+
+μ=0.06, E=0.25
+90
+time
+80
+mean escape t
+70
+60
+50
+40
+0
+0.002
+0.004
+0.006
+0.008
+0.01
+forcing0.12
+μ= 1.3E2-0.21E+0.023
+0.1
+0.08
+non-Wada
+Damping
+0.06
+0.04
+Wada
+0.02
+0
+0.2
+0.25
+0.3
+0.35
+Energyenergy, Ee ¼ 0:16, where therefore there exists three exits for the trajectories to leave the scattering region. In order to
+analyze the relationship between the external forcing, the dissipation and the associated uncertainty, we have considered
+trajectories inside the scattering region under different conditions of the perturbations and analyzed the way they escape
+outside from the scattering region. This study permitted us to compute the exit basins. We have seen that for different values
+of forcing and damping the basins could present the Wada property or not, which is directly related with the unpredictability
+of the system. We have studied, via numerical simulations, for what amount of external forcing the basins start to intermin-
+gle, enhancing the unpredictability of the system. Then, we have studied for what amount of dissipation the basins loose the
+Wada property, becoming more predictable, and repeated everything for different values of the initial energy. We think our
+results are useful to gain a better understanding on the possibility to control the unpredictability in this kind of systems,
+through the use of energy dissipation. We found that it is possible to find a minimum of the dissipation for which the basins
+are not Wada, but still compact. We have computed this minimum value for different energies and we have found a poly-
+nomial relation between the energy and the dissipation. Moreover, we have calculated, for a fixed initial energy, the pair of
+damping and forcing values, for which the basins start to show the Wada property. This analysis can be useful to know, in a
+system that presents Wada basins in phase space, where they appear in order to understand better where the system pre-
+sents more unpredictability and ways to control it.
+Acknowledgment
+We acknowledge financial support by the Spanish Ministry of Science and Innovation under Project No. FIS2009-09898.
+References
+[1] Gammaitoni L, Hänggi P, Jung P, Marchesoni F. Stochastic resonance. Rev Mod Phys 1998;70:223.
+[2] Braiman Y, Goldhirsch J. Taming chaotic dynamics with weak periodic perturbation. Phys Rev Lett 1991;66:2545.
+[3] Yang L, Liu Z, Chen G. Chaotifying a continuous via impulse input. Int J Bifurcation Chaos 2002;12:1121.
+[4] Ott E, Grebogi C, Yorke J. Controlling chaos. Phys Rev Lett 1990;64:1196.
+[5] Seoane JM, Sanjuán MAF. New developments in classical chaotic scattering. Rep Prog Phys 2013;76:016001.
+[6] Contopoulos G, Kandrup HE, Kaufman D. Fractal properties of escape from a two-dimensional potential. Physica D 1993;64:310.
+[7] Chen J, Rexford JL, Lee YC. Fractal boundaries in magnetotail particle dynamics. Geophys Res Lett 1990;17:1049.
+[8] Aguirre J, Viana RL, Sanjuán MAF. Fractal structures in nonlinear dynamics. Rev Mod Phys 2009;81:333.
+[9] Bleher S, Grebogi C, Ott E. Bifurcation to chaotic scattering. Physica D 1990;46:87.
+[10] Aguirre J, Vallejo JC, Sanjuán MAF. Wada basins and chaotic invariant sets in the Hénon–Heiles system. Phys Rev E 2001;64:066208.
+[11] Poon L, Campos J, Ott E, Grebogi C. Wada basin boundaries in chaotic scattering. Int J Bifurcation Chaos 1996;6:251.
+[12] Kovács Z, Wiesenfeld L. Topological aspects of chaotic scattering in higher dimensions. Phys Rev E 1997;63:57.
+[13] Sweet D, Ott E, Yorke J. Topology in chaotic scattering. Nature 1999;399:315.
+[14] Feudel U, Grebogi C. Multistability and the control of complexity. Chaos 1997;7:597.
+[15] Motter AE, Lai YC. Dissipative chaotic scattering. Phys Rev E 2001;65:015205R.
+[16] Kraut S, Feudel U, Grebogi C. Preference of attractors in noisy multistable systems. Phys Rev E 1999;59:5253.
+[17] Seoane JM, Aguirre J, Sanjuán MAF, Lai YC. Basin topology in dissipative chaotic scattering. Chaos 2006;16:023101.
+[18] Seoane JM, Sanjuán MAF, Lai YC. Fractal dimension in dissipative chaotic scattering. Phys Rev E 2007;76:016208.
+[19] Gan C, Yang S, Lei H. Noisy scattering dynamics in the randomly driven Hénon–Heiles oscillator. Phys Rev E 2010;82:066204.
+[20] Hénon M, Heiles C. The applicability of the third integral of motion: some numerical experiments. Astron J 1964;69:73.
+[21] Seoane JM, Sanjuán MAF. Exponential decay and scaling laws in noisy chaotic scattering. Phys Lett A 2008;372:110.
+[22] Seoane JM, Huang L, Sanjuán MAF, Lai YC. Effect of noise on chaotic scattering. Phys Rev E 2009;79:047202.
+[23] Seoane JM, Sanjuán MAF. Escaping dynamics in presence of dissipation and noise in scattering systems. Int J Bifurcation Chaos 2010;20:2783.
+[24] Chin SA, Chen CR. Forward symplectic integrators for solving gravitational few-body problems. Celestial Mech Dyn Astron 2005;91:301.
+[25] Kennedy J, Yorke JA. Basins of Wada. Physica D 1991;51:213.
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+M. Coccolo et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 3449–3457
+3457
+

JNE2T4oBgHgl3EQf_wld/content/tmp_files/2301.04251v1.pdf.txt

@@ -0,0 +1,1191 @@
+arXiv:2301.04251v1  [stat.ME]  11 Jan 2023
+On classical and Bayesian inference for bivariate Poisson conditionals
+distributions: Theory, methods and applications
+Barry C. Arnold1, Indranil Ghosh2,
+1 University of California, Riverside
+2University of North Carolina, Wilmington, USA
+Abstract
+Bivariate count data arise in several different disciplines (epidemiology, market-
+ing, sports statistics, etc., to name but a few) and the bivariate Poisson distribution
+which is a generalization of the Poisson distribution plays an important role in
+modeling such data. In this article, we consider the inferential aspect of a bivariate
+Poisson conditionals distribution for which both the conditionals are Poisson but
+the marginals are typically non-Poisson. It has Poisson marginals only in the case
+of independence. It appears that a simple iterative procedure under the maximum
+likelihood method performs quite well as compared with other numerical subrou-
+tines, as one would expect in such a case where the MLEs are not available in closed
+form. In the Bayesian paradigm, both conjugate priors and non-conjugate priors
+have been utilized and a comparison study has been made via a simulation study.
+For illustrative purposes, a real-life data set is re-analyzed to exhibit the utility of
+the proposed two methods of estimation, one under the frequentist approach and
+the other under the Bayesian paradigm.
+Keywords. Bivariate Poisson conditionals distribution; Gamma distribution mixtures;
+Maximum likelihood estimation; Bayesian estimation; Conjugate priors.
+1
+Introduction
+Bivariate count data arise in many circumstances. For example, in medicine, we may
+have pretreatment and post treatment measurements of the same individuals, or we may
+consider the incidence of two diseases in certain sites. Paired count data also arise in
+various other domains affecting our daily lives, such as economics, medical science, sports
+medicine, reliability of a production process etc. Bivariate discrete Poisson distributions
+1
+
+have enjoyed a good amount of attention over the last couple of decades or so. Various
+different versions of the bivariate Poisson distribution have been adequately discussed in
+the literature. Additionally, several different strategies to estimate the model parameters
+under both the frequentist as well as under the Bayesian paradigm have been developed.
+For a comprehensive treatment of the bivariate Poisson distribution and its multi-
+variate extensions the reader can refer to Kocherlakota and Kocherlakota (2017), and
+Johnson, Kotz, and Balakrishnan (1997).
+Below, we provide a non-exhaustive list of
+related pertinent references.
+Recently, to remedy against the problem of computational difficulties related to sta-
+tistical inference for a bivariate and multivariate Poisson distribution, many authors have
+proposed efficient and tricky strategies. Some useful references in this context can be
+cited as follows. For example, the even-points method by Papageorgiou and Kemp (1977)
+in the context of a bivariate generalized Poisson distribution; the use of conditional-even
+points method introduced by Papageorgiou and Loukas (1988) to estimate the model pa-
+rameters of a bivariate Poisson distribution can be cited as well. Holgate (1964) discussed
+the estimation of the covariance parameter for a correlated bivariate Poisson distribu-
+tion and advocated for the use of iterative method of solving the likelihood equations as
+compared to the method of moments strategy under the classical set-up. Belov (1993)
+has established the result on the uniqueness of the maximum likelihood estimates for the
+parameters of the bivariate Poisson distribution. For a Monte Carlo study concerning the
+performance of alternative estimators, see Paul and Ho (1989).
+Estimation of parameters under the Bayesian paradigm has also been developed for uni-
+variate, bivariate and multivariate Poisson distributions. For example, Karlis and Nt-
+zoufras (2006) have discussed the Bayesian analysis of the difference of count data assum-
+ing a bivariate Poisson distribution. Karlis and Tsiamyrtzis (2008) provided a framework
+to conduct an exact Bayesian analysis for bivariate Poisson data assuming conjugate
+gamma priors.
+Mo and Kockelman (2006) developed a Bayesian multivariate Poisson
+regression model useful in modeling injury count data. Tsionas (1999) has discussed the
+Bayesian analysis of the multivariate Poisson distribution based on Gibbs sampling and
+by invoking a data augmentation strategy.
+However, there has been not much discussion and study of negatively correlated bivari-
+ate Poisson distributions. Consequently, not much work has been done regarding Bayesian
+2
+
+estimating of the model parameters. This serves as one of the major motivations to carry
+out the present research work.
+In this article, we discuss the estimation (under both the frequentist and the Bayesian
+paradigm) of the model parameters of a bivariate Poisson conditionals distribution inde-
+pendently discussed by Obrechkoff (1963) and in Arnold et al. (1999). This distribution
+has also been independently discussed by Wesolowski (1996). The rest of this paper is
+organized as follows. In Section 2, we introduce the bivariate Poisson conditionals distri-
+butions due to Arnold et al. (1999) and Obrechkoff. In Section 3, we discuss the maximum
+likelihood method of estimating the model parameters via an iterative process that is dif-
+ferent from that used by Ghosh et al. (2021). Section 5 outlines Bayesian inference for the
+bivariate Poisson conditional type distributions using informative priors. The simulated
+results are presented in Section 6. Section 7 discusses the Bayesian estimation using the
+posterior mode(s) as the posterior summary for the model parameters. For illustrative
+purposes, a real-life data set has been re-analyzed to exhibit the efficacy of the proposed
+two methods of estimation under the frequentist and under the Bayesian framework in
+Section 8. Finally, some concluding remarks are provided in Section 9.
+2
+Bivariate Poisson conditionals distributions
+We begin our discussion in this section by introducing a bivariate discrete distribution for
+which both sets of conditionals are univariate Poisson according to Arnold et al. (1999)
+(p.96-97). This probability model also appears in Obrechkoff (1963) and in Wesolowski
+(1996).
+Let us assume the following:
+• X|Y = y ∼ Poisson (λ1λy
+3) , for each fixed Y = y.
+• Y |X = x ∼ Poisson (λ2λx
+3) , for each fixed X = x.
+Here, (λ1, λ2) > 0,
+0 < λ3 ≤ 1. Note that if λ3 = 1, X and Y are independent.
+According to Arnold et al. (1999) [see Theorem 4.1, page 76] the associated joint
+p.m.f. will be
+P (X = x, Y = y) = K (λ1, λ2, λ3) × λx
+1λy
+2λxy
+3
+x!y!
+,
+(2.1)
+3
+
+where x = 0, 1, 2, · · · ;
+y = 0, 1, 2, · · · , and K (λ1, λ2, λ3) is the normalizing constant
+and
+K−1 = K−1 (λ1, λ2, λ3) =
+∞
+�
+y=0
+λy
+2
+y! exp (λ1λy
+3) =
+∞
+�
+x=0
+λx
+1
+x! exp (λ2λx
+3) .
+The general assumption of Poisson conditionals forces one to have this structure.
+We will denote the bivariate Poisson distribution of the pair (X, Y ) with the p.m.f. in
+(2.1) as BPC (λ1, λ2, λ3) . Several useful structural properties of the joint p.m.f. in (2.1)
+have been discussed in Ghosh et al. (2021).
+In the next section we will focus our attention on the maximum likelihood estimation
+of the model parameters for the BPC distribution in (2.1) via a simple iterative strategy.
+The adopted strategy is different from the approach used in Ghosh et al. (2021). In
+that paper, the authors discussed maximum likelihood estimation using a copula based
+approach.
+3
+Iterative maximum likelihood estimation for the
+BPC distribution
+For a random sample of size n, the log-likelihood function of the bivariate Poisson condi-
+tionals distribution will be given by
+ℓ(λ) = −n log J(λ) + t1 log λ1 + t2 log λ2 + t3 log λ3 −
+n
+�
+i=1
+log(xi!) −
+n
+�
+i=1
+log(yi!),
+(3.1)
+where
+J(λ1, λ2, λ3) =
+∞
+�
+y=0
+λy
+2
+y! exp{λ1λy
+3} =
+∞
+�
+x=0
+λx
+1
+x! exp{λ2λx
+3}.
+From Eq. (3.1), the MLEs are obtained by taking partial derivatives w.r.t. λ1, λ2, λ3
+and setting them equal to zero.
+∂ℓ(λ)
+∂λ1
+= − n
+J(λ)
+�∂J(λ1, λ2, λ3)
+∂λ1
+�
++ t1
+λ1
+,
+(3.2)
+∂ℓ(λ)
+∂λ2
+= − n
+J(λ)
+�∂J(λ1, λ2, λ3)
+∂λ2
+�
++ t2
+λ2
+,
+(3.3)
+∂ℓ(λ)
+∂λ3
+= − n
+J(λ)
+�∂J(λ1, λ2, λ3)
+∂λ3
+�
++ t3
+λ3
+,
+(3.4)
+4
+
+where t1 = �n
+i=1 xi,
+t2 = �n
+i=1 yi,
+t3 = �n
+i=1 xiyi.
+Because of the nature of the J(λ1, λ2, λ3) function, we can rewrite the Eqs. (3.2)-(3.4) as
+J(λ1, λ2λ3, λ3)
+J(λ1, λ2, λ3)
+= t1
+nλ1
+,
+J(λ1λ3, λ2, λ3)
+J(λ1, λ2, λ3)
+= t2
+nλ2
+,
+λ1λ2J(λ1λ3, λ2λ3, λ3)
+J(λ1, λ2, λ3)
+= t3
+nλ3
+.
+It can be easily verified that the asymptotic variance-covariance of the MLEs of λ1, λ2, and
+λ3 cannot be obtained analytically because of the complicated nature of the expectations.
+Therefore, we obtain the approximate asymptotic variance-covariance matrix for the
+MLEs by getting the inverse of the observed FIM, which is as follows.
+I
+�
+�λ1, �λ2, �λ3
+�
+=
+
+
+−∂2ℓ(λ)
+∂λ2
+1
+− ∂2ℓ(λ)
+∂λ1∂λ2
+− ∂2ℓ(λ)
+∂λ1∂λ2
+∂2ℓ(λ)
+∂λ2∂λ1
+−∂2ℓ(λ)
+∂λ2
+2
+− ∂2ℓ(λ)
+∂λ2∂λ3
+− ∂2ℓ(λ)
+∂λ3∂λ1
+− ∂2ℓ(λ)
+∂λ3∂λ2
+−∂2J(λ)
+∂λ3
+3
+
+
+=
+
+
+V ar( �λ1)
+V ar( �λ2)
+V ar( �λ3)
+
+ .
+(3.5)
+The asymptotic variance-covariance matrix of the MLE λ = (ˆλ1, ˆλ2, ˆλ3) can be obtained
+from the inverse of the observed Fisher information matrix as
+V = I−1(λ)
+def.
+=
+
+
+v11
+v12
+v13
+v22
+v23
+v33.
+
+
+Under mild regularity conditions,
+�
+�λ1, �λ2, �λ3
+�
+∼ N3
+�
+(λ1, λ2, λ3), V
+�
+.
+Therefore, a 100 (1 − τ)% approximate confidence intervals of the parameters �λi will
+be
+���λi ± Z (1 − τ/2) × √vii,
+5
+
+i = 1, 2, 3, where Zq is the 100q-th upper percentile of the standard normal distribution.
+Next, in this case, the elements of of the observed FIM are:
+•
+∂2ℓ(λ)
+∂λ2
+1
+= − t1
+λ2
+1 − n
+�
+J(λ)×J(λ1,λ2λ2
+3,λ3)−(J(λ1,λ2λ3,λ3))2
+J2(λ)
+�
+.
+•
+∂2ℓ(λ)
+∂λ2
+2
+= − t2
+λ2
+2 − n
+�
+J(λ)×J(λ1λ2
+3,λ2,λ3)−(J(λ1λ3,λ2,λ3))2
+J2(λ)
+�
+.
+•
+∂2ℓ(λ)
+∂λ2
+3
+= − t3
+λ2
+3 − n
+�
+J(λ)×J(λ1λ2
+3,λ2λ2
+3,λ3)×(λ1λ2)2−(λ1λ2)×(J(λ1λ3,λ2λ3,λ3))2
+J2(λ)
+�
+.
+• Again,
+∂2ℓ(λ)
+∂λ1∂λ2
+= J
+�
+λ1λ3, λ2λ2
+3, λ3
+�
+.
+• Again,
+∂2ℓ(λ)
+∂λ1∂λ3
+= λ2J (λ1λ3, λ2λ3, λ3) + (λ1λ3) J
+�
+λ1λ3, λ2λ2
+3, λ3
+�
+.
+• Also,
+∂2ℓ(λ)
+∂λ2∂λ3
+= λ3J
+�
+λ1λ2
+3, λ2λ3, λ3
+�
+.
+Instead of using any optimization program/subroutine, we consider the following ap-
+proach to obtain the MLEs of λ1, λ2, λ3. Observe that, the above likelihood equations can
+be re-written as
+λ1 = t1J(λ1, λ2, λ3)
+nJ(λ1, λ2λ3, λ3)
+, (A)
+λ2 = t2J(λ1, λ2, λ3)
+nJ(λ1λ3, λ2, λ3),
+(B)
+λ3 =
+t3J(λ1, λ2, λ3)
+nλ1λ2J(λ1λ3, λ2λ3, λ3).
+(C)
+Next, we adopt the following simple (repetitive) process:
+• First, we pick initial values for the the three λi’s.
+6
+
+• Next, use (A) with the three current values for the λ’s on the right side to update
+λ1.
+• Next, use (B) with the three current values for the λ’s on the right side to update
+λ2.
+• Finally, use (C) with the three current values for the λ’s on the right side to update
+λ3.
+• We continue this process until the process converges in the sense that we stop at
+stage m if |λm
+i − λm+1
+i
+| < ǫ, where ǫ is a very small quantity < 0.005.
+4
+Simulation Study
+Let us assume that a random sample of size n is drawn from the joint p.m.f. in (2.1). In
+particular, we consider the sample sizes n = 50, 75 and 100 with the following four sets
+of choices of the model parameters:
+(a) Choice 1: λ1 = 2, λ2 = 2.5 and λ3 = 0.35.
+(b) Choice 2: λ1 = 1.75, λ2 = 3.25 and λ3 = 0.45.
+(c) Choice 3: λ1 = 2.5, λ2 = 1.5 and λ3 = 0.55.
+(d) Choice 4: λ1 = 3.5, λ2 = 4 and λ3 = 0.75.
+Random samples from the BPC distribution are generated using the techniques discussed
+in Shin and Pasupathy (2010). The MLEs of λ1, λ2, and λ3 are obtained by adopting the
+strategy described in the previous section.
+For each of these choices above, the following initial values of the parameters are
+considered
+(a) Initial values for Choice 1: λ1 = 1.04, λ2 = 1.23 and λ3 = 0.125.
+(b) Initial values for Choice 2: λ1 = 0.98, λ2 = 1.46 and λ3 = 0.27.
+(c) Initial values for Choice 3: λ1 = 1.12, λ2 = 1.03 and λ3 = 0.28.
+(d) Initial values for Choice 4: λ1 = 1.74, λ2 = 2.23 and λ3 = 0.18.
+7
+
+Table 4.1: Simulated coverage probabilities (CP) and average widths (AW) of the MLEs of the parameters in the BPCN distribution for
+various choices of λ
+Parameter choice
+Based on asymptotic variances from inverting I(λ)
+Based on bootstrap variances
+λ1
+λ2
+λ3
+% of negative
+λ1
+λ2
+λ3
+n
+CP
+AW
+CP
+AW
+CP
+AW
+variances
+CP
+AW
+CP
+AW
+CP
+AW
+Choice 1
+50
+0.950
+1.377
+0.952
+0.563
+0.992
+2.361
+0.0475
+0.913
+1.388
+0.938
+0.535
+0.922
+1.732
+75
+0.939
+1.243
+0.953
+0.487
+0.986
+1.261
+0.090
+0.814
+1.344
+0.938
+0.485
+0.917
+1.534
+100
+0.935
+1.137
+0.959
+0.432
+0.982
+0.584
+0.120
+0.923
+1.299
+0.935
+0.452
+0.927
+1.335
+Choice 2
+50
+0.905
+1.444
+0.940
+0.577
+0.997
+2.569
+0.130
+0.912
+1.508
+0.950
+0.570
+0.958
+1.137
+75
+0.882
+1.292
+0.943
+0.498
+0.993
+1.189
+0.170
+0.940
+1.422
+0.945
+0.508
+0.959
+1.032
+100
+0.853
+1.203
+0.943
+0.448
+0.988
+1.032
+0.090
+0.925
+1.362
+0.944
+0.469
+0.954
+0.812
+Choice 3
+50
+0.950
+1.371
+0.949
+0.561
+0.993
+2.481
+0.110
+0.918
+1.392
+0.943
+0.535
+0.926
+1.643
+75
+0.941
+1.229
+0.956
+0.484
+0.986
+1.264
+0.170
+0.904
+1.345
+0.935
+0.486
+0.914
+1.345
+100
+0.930
+1.117
+0.955
+0.428
+0.978
+1.172
+0.110
+0.911
+1.292
+0.934
+0.449
+0.924
+0.733
+Choice 4
+50
+0.904
+1.437
+0.940
+0.575
+0.997
+2.556
+0.130
+0.936
+1.478
+0.945
+0.564
+0.955
+1.542
+75
+0.882
+1.289
+0.943
+0.496
+0.993
+1.218
+0.090
+0.935
+1.406
+0.944
+0.504
+0.953
+1.046
+100
+0.853
+1.199
+0.943
+0.447
+0.988
+1.043
+0.100
+0.921
+1.373
+0.947
+0.474
+0.947
+0.938
+8
+
+One may observe from Table 4.1, that the estimated MSEs for the three parameters
+λ1, λ2 and λ3 decrease as the sample size increases. However, for the estimated biases,
+there is not a steady decreasing pattern with the increase of sample sizes, and on the
+contrary, in some cases, it appears that there is a negligible amount (by 0.01 − 0.05) of
+increase. We observe that the direction of the estimated biases of the MLE of λ3 is the
+same as the sign of the true value of the parameter λ3. Moreover, the estimated MSEs of
+λ3 is larger than the MSEs of λ1 and λ2.
+Additionally, from Table 4.1, one may also observe the following
+• that the proportions of cases in which negative variance estimates are obtained is
+negligibly small.
+• Additionally, the computed approximate confidence intervals based on bootstrap
+variances performs satisfactorily well. Note that these approximate confidence in-
+tervals can be used as an alternative when the asymptotic variances are negative
+(for pertinent details, see Ghosh and Ng (2019) and the references cited therein).
+5
+Bayesian inference
+Since the classical methods of estimation for bivariate discrete probability models does
+not always yield satisfactory results due to several factors, such as likelihood involving
+ubiquitous normalizing constants, non-existence of efficient algorithms to obtain global
+maximums for the model parameter(s) as opposed to local maximums, etc., it is legiti-
+mate to consider a Bayesian approach in this context. There are several advantages of
+conducting a Bayesian analysis, especially for bivariate discrete probability models (for
+pertinent details, see Berm´udez, L., & Karlis, D. (2011). In this section, we begin our
+discussion on the Bayesian estimation by assuming the conjugate prior set-up at first for
+the joint p.m.f. as given in Eq. (2.1). In this case, we are dealing with a three parameter
+exponential family, so a conjugate prior will exist. First we reparametrize by defining new
+parameters as follows.
+δi = log λi,
+i = 1, 2, 3.
+Note that δ1, δ2 ∈ (−∞, ∞) while δ3 ∈ (−∞, 0].
+The BPC joint p.m.f. in Eq. (2.1) can be re-written as
+9
+
+f
+�
+x, y;⃗δ
+�
+=
+�K(δ1, δ2, δ3) exp[δ1x + δ2y + δ3xy]
+x!y!
+,
+(5.1)
+where x and y are non-negative integers, and ⃗δ = (δ1, δ2, δ3) .
+The associated likelihood function corresponding to a sample of size n will then be
+L(δ) = [ �K(δ1, δ2, δ3)]n exp[δ1
+� xi + δ2
+� yi + δ3
+� xiyi]
+� xi! � yi!
+.
+(5.2)
+As a conjugate prior, one may consider the following
+fη(δ) ∝ [ �K(δ1, δ2, δ3)]η0 exp[η1δ1 + η2δ2 + η3δ3].
+(5.3)
+The corresponding posterior density will be of the same form, with adjusted hyperpa-
+rameters, i.e.,
+f(δ|t) ∝ [ �K(δ1, δ2, δ3)]η0+n exp[(η1 + t1)δ1 + (η2 + t2)δ2 + (η3 + t3)δ3],
+(5.4)
+where t1 = � xi, t2 = � yi and t3 = � xiyi.
+However, in the process of selecting the values of the hyperparameters, we note that
+the posterior density is proportional to the likelihood of a sample of size nP = η0 + n)
+with sufficient statistics ti,P = ηi + ti, i = 1, 2, 3.
+Now, in order to make a sensible choice for the four hyperparameters, we will rely on
+the fact that our informed expert has had past experience with data very similar to the
+current data set. We ask him for a typical value for observed X’s which will be denoted by
+v1, a typical value for the Y ’s to be denoted by v2 and a typical value for the XY ’s to be
+denoted by v3. Then we ask for a number or index to indicate how confident he is about
+the three typical values. Denote this by n∗. This can alternatively be viewed as being
+a consequence of having observed an “imaginary” sample of size n∗ with corresponding
+sufficient statistics
+n∗
+�
+i=1
+xi = n∗v1,
+n∗
+�
+i=1
+yi = n∗v2,
+n∗
+�
+i=1
+xiyi = n∗v3.
+Based on this information we choose as our four hyperparameters η0 = n∗, ηi =
+n∗vi,
+i = 1, 2, 3.
+We can rewrite the posterior density as a function of original λi’s, if we wish. If we
+do so, it will be the same as the log-likelihood given in Eq. (5.4) with suitably revised
+values for n, t1, t2 and t3. So, if we decide to use the posterior mode to estimate the λi’s,
+we can apply our iterative scheme to find the location of the mode.
+10
+
+6
+A simulation study
+Let us assume that the confidence index provided by our informed expert is n∗ = 12, a
+small value, indicating that the expert is not at all sure about the values v1 = 5, v2 = 4,
+v3 = 6, that are provided. Then, as per the suggestion made earlier, we have the following
+suggested values for the hyperparameters η0 = 5 η1 = 60, η2 = 48, η2 = 72. Next, the
+marginal posteriors can be obtained as (proportional to)
+•
+Π1
+�
+δ1|⃗t∗�
+∝ exp[(η1+t1)δ1]
+� ∞
+−∞
+� 0
+−∞
+[ �K(δ1, δ2, δ3)]η0+n exp [(η2 + t2)δ2 + (η3 + t3)δ3] dδ2dδ3.
+•
+Π2
+�
+δ2|⃗t∗�
+∝ exp[(η2+t2)δ2]
+� ∞
+−∞
+� 0
+−∞
+[ �K(δ1, δ2, δ3)]η0+n exp [(η1 + t1)δ1 + (η3 + t3)δ3] dδ1dδ3.
+•
+Π3
+�
+δ3|⃗t∗�
+∝ exp[(η3+t3)δ3]
+� ∞
+−∞
+� ∞
+−∞
+[ �K(δ1, δ2, δ3)]η0+n exp [(η1 + t1)δ1 + (η2 + t2)δ2] dδ1dδ2.
+If instead we wish to use the posterior expectations of the λi’s as our estimates we
+will need to use numerical integration as follows.
+• For λ1,
+E (λ1|t) =
+� ∞
+−∞
+� ∞
+−∞
+� 0
+−∞ eδ1[ �K(δ1, δ2, δ3)]η0+n exp[(η1 + t1)δ1 + (η2 + t2)δ2 + (η3 + t3)δ3] dδ1 dδ2 dδ3
+� ∞
+−∞
+� ∞
+−∞
+� 0
+−∞[ �K(δ1, δ2, δ3)]η0+n exp[(η1 + t1)δ1 + (η2 + t2)δ2 + (η3 + t3)δ3] dδ1 dδ2 dδ3
+.
+• For λ2,
+E (λ2|t) =
+� ∞
+−∞
+� ∞
+−∞
+� 0
+−∞ eδ2[ �K(δ1, δ2, δ3)]η0+n exp[(η1 + t1)δ1 + (η2 + t2)δ2 + (η3 + t3)δ3] dδ1 dδ2 dδ3
+� ∞
+−∞
+� ∞
+−∞
+� 0
+−∞[ �K(δ1, δ2, δ3)]η0+n exp[(η1 + t1)δ1 + (η2 + t2)δ2 + (η3 + t3)δ3] dδ1 dδ2 dδ3
+.
+• For λ3,
+E (λ3|t) =
+� 0
+−∞
+� ∞
+−∞
+� ∞
+−∞ eδ3[ �K(δ1, δ2, δ3)]η0+n exp[(η1 + t1)δ1 + (η2 + t2)δ2 + (η3 + t3)δ3] dδ1 dδ2 dδ3
+� ∞
+−∞
+� ∞
+−∞
+� 0
+−∞[ �K(δ1, δ2, δ3)]η0+n exp[(η1 + t1)δ1 + (η2 + t2)δ2 + (η3 + t3)δ3] dδ1 dδ2 dδ3
+.
+11
+
+Note that higher order moments can also be obtained (via numerical methods, of course).
+The choice of priors will have a significant impact on both bias and computational time.
+We consider the posterior mean as the Bayes estimates for the parameters.
+We also
+provide 95% credible intervals as a summary related to Bayesian estimation that are
+given in Table 6.1. We consider the following four different parameter settings:
+• Choice 1: δ1 = −2.5;
+δ2 = −1.3;
+δ3 = −0.25.
+• Choice 2: δ1 = −1.8;
+δ2 = −0.98;
+δ3 = −0.45.
+• Choice 3: δ1 = 1.58;
+δ2 = 1.87;
+δ3 = −0.55.
+• Choice 4: δ1 = 0.46;
+δ2 = 0.92;
+δ3 = −0.65.
+6.1
+Bayesian analysis with locally uniform priors
+In this case, we consider the following locally uniform priors for the three parameters
+which are as follows:
+• Π(δ1) ∝ 1,
+for
+− ∞ < δ1 < ∞.
+• Π(δ2) ∝ 1,
+for
+− ∞ < δ2 < ∞,
+and
+• Π(δ3) ∝ 1, −∞ < δ3 < 0.
+If we, before observing the imaginary sample, assume that the parameters had a flat joint
+prior (that are given above), then the posterior, after observing the imaginary sample
+would be just the likelihood of the imaginary sample, i.e.,
+L∗(δ) ∝ [ �K(δ1, δ2, δ3)]n∗ exp [δ1n∗v1 + δ2n∗v2 + δ3n∗v3] .
+(6.1)
+It is this posterior that we will use for a prior for the real data set. Therefore, the
+resulting posterior combining the data likelihood given in Eq. (5.2) with the prior given
+in Eq. (6.2) will be
+Π
+�
+δ|⃗v,⃗t
+�
+∝ [ �K(δ1, δ2, δ3)]n∗+n exp [δ1 (n∗v1 + t1) + δ2 (n∗v2 + t2) δ3 (n∗v3 + t3)] .
+(6.2)
+Subsequently, the posterior means for the parameters are obtained which will be
+12
+
+Table 6.1: Posterior summary for the BPC model under the conjugate prior assumption
+Parameter choices
+�
+λ1
+�
+λ2
+�
+λ3
+Posterior mean
+95% HPD
+Posterior mean
+95% HPD
+Posterior mean
+95% HPD
+Choice 1
+0.0751
+(0.0386, 2.9921)
+0.2611
+(0.0767, 1.8154)
+0.7718
+( 0.5018, 0.8541)
+Choice 2
+0.1725
+(0.1277, 1.0568)
+0.3598
+(0.1429, 1.3422)
+0.6389
+(0.3479,0.6817)
+Choice 3
+4.8695
+(1.076, 6.3756)
+1.932
+(1.1921, 3.8764)
+0.5521
+( 0.5040, 0.9483)
+Choice 4
+1.5688
+(1.389, 5.0218)
+2.487
+(1.597, 3.4856)
+0.5127
+(0.4082, 0.7527)
+13
+
+•
+E
+�
+λ1|⃗v,⃗t
+�
+=
+� ∞
+−∞
+� ∞
+−∞
+� 0
+−∞[ �K(δ1, δ2, δ3)]n∗+n
+�
+exp [δ1 (n∗v1 + t1 + 1) δ2 (n∗v2 + t2) + δ3 (n∗v3 + t3)]
+�
+dδ1dδ2dδ3
+� � ∞
+−∞
+� ∞
+−∞
+� 0
+−∞[ �K(δ1, δ2, δ3)]n∗+n exp [δ1 (n∗v1 + t1) + δ2 (n∗v2 + t2) δ3 (n∗v3 + t3)] dδ1dδ2dδ3
+�
+•
+E
+�
+λ2|⃗v,⃗t
+�
+=
+� ∞
+−∞
+� ∞
+−∞
+� 0
+−∞[ �K(δ1, δ2, δ3)]n∗+n
+�
+exp [δ1 (n∗v1 + t1) δ2 (n∗v2 + t2 + 1) + δ3 (n∗v3 + t3)]
+�
+dδ1dδ2dδ3
+� ∞
+−∞
+� ∞
+−∞
+� 0
+−∞[ �K(δ1, δ2, δ3)]n∗+n exp [δ1 (n∗v1 + t1) + δ2 (n∗v2 + t2) δ3 (n∗v3 + t3)] dδ1dδ2dδ3
+.
+•
+E
+�
+λ3|⃗v,⃗t
+�
+=
+� ∞
+−∞
+� ∞
+−∞
+� 0
+−∞[ �K(δ1, δ2, δ3)]n∗+n
+�
+exp [δ1 (n∗v1 + t1) δ2 (n∗v2 + t2) + δ3 (n∗v3 + t3 + 1)]
+�
+dδ1dδ2dδ3
+� ∞
+−∞
+� ∞
+−∞
+� 0
+−∞[ �K(δ1, δ2, δ3)]n∗+n exp [δ1 (n∗v1 + t1) + δ2 (n∗v2 + t2) δ3 (n∗v3 + t3)] dδ1dδ2dδ3
+.
+In this case, we use the same set of four different choices of the model parameters listed
+earlier for the simulation study.
+7
+Bayesian inference using posterior mode
+If we re-write the joint posterior in Eq. (2.4) in terms of the original λ′
+i s, the expression
+will be [under the conjugate prior set-up]
+f(λ|t) ∝ [K(λ1, λ2, λ3)]η0+nλη1+t1
+1
+λη2+t2
+2
+λη3+t3
+3
+.
+(7.1)
+Next, it is straightforward to to find the posterior mode of (λ1, λ2, λ3) using Newton-
+Raphson and to obtain approximate posterior standard deviations of (λ1, λ2, λ3) using the
+second derivative matrix of the log posterior evaluated at the mode. However, we only
+report the posterior mode values.
+For the simulation study, we select the same set of parameter choices as in the case of
+MLE. A random sample of size n = 100 is drawn from the joint distribution.
+(a) Choice 1: λ1 = 2, λ2 = 2.5 and λ3 = 0.35.
+14
+
+Table 6.2: Posterior summary for the BPC model under the non-informative prior assumption
+Parameter choices
+�
+λ1
+�
+λ2
+�
+λ3
+Posterior mean
+95% HPD
+Posterior mean
+95% HPD
+Posterior mean
+95% HPD
+Choice 1
+0.0725
+(0.03148, 2.8066)
+0.2432
+(0.0831, 7.7160)
+0.7795
+( 0.5204, 0.8503)
+Choice 2
+0.1676
+(0.1125, 3.2384)
+0.3113
+(0.1316, 2.3809)
+0.6492
+(0.2841, 0.6756)
+Choice 3
+4.9258
+(2.7871, 11.2054)
+6.6925
+(2.8171, 9.4934)
+0.5559
+( 0.4904, 0.9525)
+Choice 4
+3.2546
+(1.295, 5.2314)
+3.485
+(1.4782, 5.4218)
+0.5071
+(0.3929, 0.7459)
+15
+
+Table 7.1: Posterior modes for the BPC model under the conjugate prior assumption
+Parameter choices
+�
+λ1
+�
+λ2
+�
+λ3
+Posterior mode
+Posterior mode
+Posterior mode
+Choice 1
+2.131
+2.522
+0.315
+Choice 2
+1.784
+1.5783
+1.467
+Choice 3
+2.539
+1.487
+0.583
+Choice 4
+3.601
+3.926
+0.743
+(b) Choice 2: λ1 = 1.75, λ2 = 3.25 and λ3 = 0.45.
+(c) Choice 3: λ1 = 2.5, λ2 = 1.5 and λ3 = 0.55.
+(d) Choice 4: λ1 = 3.5, λ2 = 4 and λ3 = 0.75.
+For the conjugate prior set-up, we consider the following values of the hyperparameters:
+η0 = 1.23,
+η1 = 2.325,
+η2 = 3.25,
+η3 = 2.528. We report the location of the posterior
+modes of the posterior as a summary related to Bayesian estimation that are given in
+Table 7.1.
+8
+Real-data application
+To illustrate the feasibility of the proposed two Bayesian approaches in the preceding sec-
+tion, we consider the data which is originally due to Aitchison and Ho (1989). This data
+set has also been studied independently by Lee et al. (2017) and Ghosh et al. (2021). For
+pertinent details on this particular data set and the applicability of the bivariate Poisson
+conditionals distribution as a reasonable fit for this data set, see Ghosh et al. (2021). In
+this subsection, we re-analyze this dataset under the Bayesian paradigm.
+Next, for the Bayesian analysis, we make a note of the following:
+• For the conjugate prior set-up, we consider the following values of the hyperparam-
+eters: η0 = 1.41,
+η1 = 2.325,
+η2 = 3.25,
+η3 = 2.528.
+• For the locally uniform prior set-up, we consider the joint prior as given in Eq. (6.1).
+The parameter estimates (posterior mean, highest posterior density interval) under both
+the conjugate prior and the locally uniform priors are provided in Table 8.1.
+16
+
+Table 8.1: Goodness of fit summary for the Lens data under the BPC model )
+Parameter choices
+�
+λ1
+�
+λ2
+�
+λ3
+Posterior mean
+95% HPD
+Posterior mean
+95% HPD
+Posterior mean
+95% HPD
+Conjugate prior set-up
+1.8500
+(1.3832, 3.6052)
+2.1699
+(1.7633, 6.400)
+0.9600
+( 0.5574, 0.9832)
+Locally uniform prior set-up
+1.8650
+(1.2926, 4.1516)
+2.1878
+(1.6507,6.8579)
+0.9574
+(0.3378, 1.0211)
+17
+
+From the Table 6.1, it appears that the parameter estimates obtained with the conjugate
+prior choice closely matches the MLE estimates obtained using the copula as discussed
+in Ghosh et al. (2021). Under the flat prior set-up, the length of 95% HPD intervals are
+slightly wider as can be observed from Table 8.1, second row—third, fifth and the seventh
+column values.
+9
+Conclusion
+Modeling of bivariate paired count data is an open problem because of the inadequate class
+of bivariate discrete distributions, which if available, might explain the true dependence
+structure effectively. In this paper, we focus on the classical (using an iterative approach)
+and Bayesian inference for a bivariate discrete probability distribution for which both the
+conditionals belong to an univariate Poisson distribution with appropriate parameters,
+and the distribution is described by Arnold et al. (1999), which will always have negative
+correlation, except in the independent case. In this paper, we have discussed an alternative
+iterative algorithm for the maximum likelihood method under the frequentist set-up which
+has a striking advantage that we don’t need any maximizing/optimizing root finding
+subroutines and which can be implemented in any programming environment via some
+user defined package(s). On the Bayesian inferential aspect, both the conjugate and the
+locally uniform prior set-up have been assumed. While a conjugate prior set-up is quite
+natural for the joint p.m.f. of the form as given in (5.1), it is challenging to find a conjugate
+prior in such a scenario from a real-world perspective. A full scale study under both the
+classical and Bayesian paradigm for a multivariate Poisson conditional distribution can
+be considered from a real-life perspective where such a model will be useful. We did not
+pursue this problem here as it is beyond the scope of this paper.
+Disclosure Statement
+The authors do not have a competing interest.
+18
+
+References
+[1] Arnold, B.C., Castillo, E., and Sarabia, J.M. (1999). Conditional Specification of
+Statistical Models. Springer, New York.
+[2] Aitchison, J., & Ho, C. H. (1989). The multivariate Poisson-log normal distribution.
+Biometrika, 76(4), 643-653.
+[3] Aktekin, T., Polson, N., & Soyer, R. (2018). Sequential Bayesian analysis of multi-
+variate count data. Bayesian Analysis, 13(2), 385-409.
+[4] Belov, A. G. (1993). On the uniqueness of maximum likelihood estimates for the
+parameters of the bivariate Poisson distribution. Vestnik MGU, Series 15, 58-59 (in
+Russian).
+[5] Brooks, S., Gelman, A., Jones, G., & Meng, X. L. (Eds.). (2011). Handbook of
+Markov chain Monte Carlo. CRC press.
+[6] Berm´udez, L., & Karlis, D. (2011). Bayesian multivariate Poisson models for insur-
+ance ratemaking. Insurance: Mathematics and Economics, 48(2), 226-236.
+[7] Chib, S., & Winkelmann, R. (2001). Markov chain Monte Carlo analysis of correlated
+count data. Journal of Business & Economic Statistics, 19(4), 428-435.
+[8] Ghosh, I., Marques, F., & Chakraborty, S. (2021). A new bivariate Poisson distri-
+bution via conditional specification: properties and applications. Journal of Applied
+Statistics, 48(16), 3025-3047.
+[9] Ghosh, I., & Ng, H. K. T. (2019). A class of skewed distributions with applications
+in environmental data. Communications in Statistics: Case Studies, Data Analysis
+and Applications, 5(4), 346-365.
+[10] Holgate, P. (1964). Estimation for the bivariate Poisson distribution. Biometrika,
+51(1-2), 241-287.
+[11] Johnson, N. L., Kotz, S., & Balakrishnan, N. (1997). Discrete multivariate distribu-
+tions (Vol. 165). New York: Wiley.
+[12] Karlis, D., & Ntzoufras, I. (2006). Bayesian analysis of the differences of count data.
+Statistics in medicine, 25(11), 1885-1905.
+19
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+[13] Karlis, D., & Tsiamyrtzis, P. (2008). Exact Bayesian modeling for bivariate Poisson
+data and extensions. Statistics and Computing, 18(1), 27-40.
+[14] Kocherlakota, S., & Kocherlakota, K. (2017). Bivariate discrete distributions. CRC
+Press.
+[15] Lee, H., Cha, J. H., & Pulcini, G. (2017). Modeling discrete bivariate data with appli-
+cations to failure and count data. Quality and Reliability Engineering International,
+33(7), 1455-1473.
+[16] Loukas, S., & Kemp, C. D. (1986). The index of dispersion test for the bivariate
+Poisson distribution. Biometrics, 941-948.
+[17] Ma, J., & Kockelman, K. M. (2006). Bayesian multivariate Poisson regression for
+models of injury count, by severity. Transportation Research Record, 1950(1), 24-34.
+[18] Papageorgiou, H., & Kemp, C. D. (1977). Even point estimation for bivariate gener-
+alized Poisson distributions. Statistical Reports and Preprints, (29).
+[19] Papageorgiou, H., & Loukas, S. (1988). Conditional even point estimation for bi-
+variate discrete distributions. Communications in Statistics-Theory and Methods,
+17(10), 3403-3412.
+[20] Paul, S. R., & Ho, N. I. (1989). Estimation in the bivariate Poisson distribution and
+hypothesis testing concerning independence. Communications in Statistics-Theory
+and Methods, 18(3), 1123-1133.
+[21] Shin, K., & Pasupathy, R. (2010). An algorithm for fast generation of bivariate
+Poisson random vectors. INFORMS Journal on Computing, 22(1), 81-92.
+[22] Obrechkoff, N. (1963). Theory of Probability. Nauka i Izkustvo, Sofia.
+[23] Tsionas, E. G. (1999). Bayesian analysis of the multivariate Poisson distribution.
+Communications in Statistics-Theory and Methods, 28(2), 431-451.
+[24] Wesolowski, J. (1996). A new conditional specification of the bivariate Poisson con-
+ditionaIs distribution. Statistica Neerlandica, 50(3), 390-393.
+20
+

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+ 
+Optical-controlled ultrafast dynamics of skyrmion in antiferromagnets  
+ 
+S. H. Guan1, Y. Liu1, Z. P. Hou1, D. Y. Chen1, Z. Fan1, M. Zeng1, X. B. Lu1, X. S. Gao1,  
+M. H. Qin1,*, and J. –M. Liu1,2 
+1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials 
+and Institute for Advanced Materials, South China Academy of Advanced Optoelectronics, 
+South China Normal University, Guangzhou 510006, China 
+2Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China 
+ 
+[Abstract] Optical vortex, a light beam carrying orbital angular momentum (OAM) has been 
+realized in experiments, and its interactions with magnets show abundant physical 
+characteristics and great application potentials. In this work, we propose that optical vortex 
+can control skyrmion ultrafast in antiferromagnets using numerical and analytical methods. 
+Isolated skyrmion can be generated/erased in a very short time ~ps by beam focusing. 
+Subsequently, the OAM is transferred to the skyrmion and results in its rotation motion. 
+Different from the case of ferromagnets, the rotation direction can be modulated through 
+tuning the light frequency in antiferromagnets, allowing one to control the rotation easily. 
+Furthermore, the skyrmion Hall motion driven by multipolar spin waves excited by optical 
+vortex is revealed numerically, demonstrating the dependence of the Hall angle on the OAM 
+quantum number. This work unveils the interesting optical-controlled skyrmion dynamics in 
+antiferromagnets, which is a crucial step towards the development of optics and spintronics.  
+ 
+Keywords: optical vortex, orbital angular momentum, skyrmion, antiferromagnets 
+ 
+Email: [email protected] 
+
+Typically, photons can possess both linear momentum along the propagation direction 
+and spin angular momentum (SAM) related to circular polarization or chirality [1-3]. 
+Interestingly, as a new type of beam carrying orbit angular momentum (OAM), optical vortex 
+was predicted [4-8] and experimentally realized through using optical elements such as the 
+computer-generated holograms, mode conversions, and spiral phase plates [9-11]. Generally, 
+optical vortex has a helical phase wave front which is characterized by an azimuthal phase 
+factor eim with the OAM quantum number m. Moreover, the beam forms a ring-shaped 
+spatial profile of the intensity in the cross section, attributing to zero field topological 
+singularity in vortex core [12,13].  
+Due to its abundant physical connotation, optical vortex and its potential applications 
+have received extensive attention. For instance, it has been suggested to be used as 
+super-resolution microscopy [14] and chiral laser ablation [15,16]. Moreover, it may play an 
+important role in modulating particle dynamics through OAM transfer. Specifically, the OAM 
+transfer from optical vortex to particles drives the rotation of the latter around the beam axis, 
+because that the rotating energy flux induced by Poynting vector propagation exerts a torque 
+on the particles [17-19]. Thus, optical vortex could be used in optical micromachines like 
+optically driven cogs [19]. Interestingly, besides classical particles, quasiparticles such as 
+magnetic skyrmion [20-25] can also be controlled by optical vortex through OAM transfer, 
+which is significantly attractive in spintronic applications.   
+Magnetic skyrmion is a topological soliton with noncollinear structure, which can be 
+stabilized by Dzyaloshinskii-Moriya interaction (DMI) in non-centrosymmetric crystals or 
+frustrated exchange interaction [26-29]. Attributing to its attractive characteristics including 
+nanoscale in size, topology protection, and low threshold driving current [30,31], skyrmion is 
+considered as a promising candidate as information carrier for future spintronic devices. Thus, 
+ultrafast manipulation of skyrmion is one of the most important topics in current spintronics. 
+Luckily, optical vortex has been revealed to be potential stimulus in fast modulating magnetic 
+skyrmions. For example, the optical vortex couples with ferromagnet through Zeeman effect 
+and induces twisted magnons, which in turn contributes to the stabilization of the skyrmions 
+[16,32]. Furthermore, the vortex beam transfers its OAM to the skyrmions, and drives the 
+skyrmions rotation around the beam axis [33]. 
+
+On the other hand, antiferromagnetic (AFM) skyrmions are drawing more and more 
+attention [34-36], considering that they are free of several disadvantages of ferromagnets 
+including the strong stray field and relatively slow spin dynamics. Unlike skyrmion in 
+ferromagnets, AFM skyrmion is comprised of two coupled spin configurations with opposite 
+topological numbers, resulting in strong anti-interference capability [37,38]. Besides, AFM 
+skyrmion exhibits more abundant physics and desirable features, such as terahertz oscillation 
+frequency and ultrafast dynamics [39,40]. 
+Undoubtedly, the manipulation of AFM skyrmion using optical vortex is an attractive 
+topic which deserves to be urgently explored based on the following aspects. First, ultrafast 
+generation of AFM skyrmion could be realized by applying optical vortex, noting that AFM 
+dynamics is generally much faster than ferromagnetic one. Second, abundant physical 
+phenomena induced by the coupling of optical vortex and AFM skyrmion are expected, 
+considering the strong antiferromagnetic exchange coupling. For example, an additional time 
+inversion symmetry term will be introduced in spin dynamic equation in antiferromagnets 
+[41], resulting in the dynamic behaviors rather different from those in ferromagnets. At last, 
+optical vortex can excite multipolar spin waves [32], whose interaction with AFM skyrmions 
+determines the dynamics of the latter. However, the multipolar spin-wave-driven dynamics of 
+AFM skyrmion remains ambiguous, although effective control of the AFM skyrmion by plane 
+spin-waves has been uncovered [42].  
+In this work, we study the manipulation of the AFM skyrmion under the stimulation of an 
+optical vortex, using numerical and analytical methods. It will be demonstrated that isolated 
+skyrmion can be generated/erased in a short time of ~ps by the optical vortex. Subsequently, 
+the OAM transfer results in the skyrmion rotation whose direction also depends on the light 
+frequency, in addition to the OAM quantum number. This interesting property allows one to 
+modulate the skyrmion dynamics easily through tuning the light frequency. Furthermore, a 
+skyrmion Hall motion driven by the vortex-induced multipolar spin waves is revealed, and the 
+Hall angle depends on both the light handedness and the OAM quantum number.     
+We consider a two-dimensional classical Heisenberg model in xy plane with the following 
+Hamiltonian, 
+i
+i
+z
+i
+j
+i
+j
+i
+j
+j
+i
+i
+m
+K
+J
+H
+m
+B
+m
+m
+D
+m
+m
+
+−
+−
+
+
++
+
+=
+
+
+
+
+
+
+
+2
+,
+,
+)
+(
+)
+(
+, 
+(1) 
+
+where mi = −Si/ћ is the local magnetic moment at site i with the local spin Si and the reduced 
+Planck constant ћ. The first term is the AFM exchange energy between the nearest neighbors 
+with J > 0, the second term is the bulk DMI with the vector D = Deij, eij is the unit vector 
+connecting the nearest neighbors, the third term is the anisotropy energy along the z-axis, and 
+the last term is the Zeeman coupling of the local magnetic moments and external field B. 
+Without loss of generality, we take the lattice constant a = 0.5 nm, the magnetic layer 
+thickness d = 2 nm, the coupling constants J = 1.0  10-21 J, D/J = 0.073, and K/J = 0.01 [36]. 
+The magnetic dynamics is described by solving the Landau-Lifshitz-Gilbert (LLG) 
+equation, 
+t
+t
+i
+i
+eff
+i
+i
+i
+d
+d
+d
+d
+m
+m
+H
+m
+m
+
++
+
+−
+=
+
+
+, 
+(2) 
+where Heff i = −(1/μ0)∂H/∂mi is the effective field, α = 0.01 is the Gilbert damping coefficient, 
+and γ = -2.211 × 105 m/(A · s) is the gyromagnetic ratio [36]. 
+Following the earlier works, a vortex beam with Laguerre-Gaussian mode is considered, 
+which carriers the following magnetic field profile at the focal plane [32,33], 
+p
+m
+p
+t
+m
+i
+t
+t
+W
+m
+W
+L
+e
+W
+W
+B
+t
+e
+B
+)
+/
+2
+(
+)
+/
+(
+)
+,
+,
+(
+2
+2
+|
+|
+)
+(
+)
+(
+2
+/
+1
+|
+|
+0
+2
+0
+2
+2
+
+
+
+
+
+
+
+
+−
++
+−
+−
+−
+−
+=
+, 
+(3) 
+where B0 is the strength of the magnetic field, W represents the beam waist,  is the distance 
+from the reference point to the vortex core, t is the relaxation time, t0 determines the peak 
+position of beam intensity, and σ, m, ,  are beam duration, OAM quantum number, polar 
+angle, and light frequency, respectively, ep = x +/− iy corresponds to the left-/right- handed 
+circularly polarized light. For the case of radial index p = 0, the generalized Laguerre function 
+Lp|m|(22/W2) = 1. Fig. 1 depicts the coupling principle of the optical vortex and the studied 
+magnetic system, where the red ball represents a photon whose SAM leads to a local 
+magnetization procession, and the OAM induces twisted magnons. Unless stated elsewhere, 
+we set B0 = 0.25 which corresponds to 1.1 Tesla, W = 5a, m = −8, σ = 1.5 ps, and  = 4 THz 
+for the optical vortex. It is worth noting that the beam in subwavelength scale can be realized 
+owing to the development of plasma technology [43].  
+First, we investigate the stabilization of the skyrmion depending on the optical vortex. Fig. 
+2(a) shows the evolution of spin configuration for m = −8, which demonstrates the effective 
+
+skyrmion writing by the optical vortex. The vortex induces twisted magnons in AFM system 
+at t = 2 ps, and the magnons are coupled due to the presence of the DMI, forming a 
+ring-shaped structure (t = 4 ps). Subsequently, excessive energy makes the structure evolve 
+into an unstable skyrmionium at t = 6 ps, which degenerates into a stable skyrmion at last (t = 
+20 ps). Similar writing processes are observed for other minus m with the writing time of ~ps, 
+which is much faster than that in ferromagnets [32]. Importantly, the writing process here is 
+independent of device geometry and is much faster than those of traditional methods. As a 
+comparison, the skyrmion writing using current pulse [44] and local heating [45] are usually 
+in the nanosecond time range. Furthermore, the skyrmion can be easily erased by reversing 
+the sign of m [33,46] through modulating the coupling of chirality and the OAM, as shown in 
+Fig. 2(b), further demonstrating the great potential of the optical vortex in manipulating AFM 
+skyrmions. 
+Then, we investigate the OAM transfer from the beam to the skyrmion to study the 
+optical driven skyrmion dynamics. The Lagrangian density can be written as [47],  
+n
+B
+n
+n
+n
+n
+n
+n
+n
+B
+n
+
+
+−
+−
+
+
+
++
+
+−
+−
+
+
+
++
+
+−
+=
+0
+2
+0
+2
+0
+2
+2
+0
+2
+2
+)
+(
+)
+(
+2
+/
+)
+(
+)
+(
+2
+A
+SL
+Kn
+A
+L
+A
+L
+A
+D
+A
+z
+
+
+
+
+
+
+
+
+, 
+(4) 
+where n = (m1 − m2)/|m1 − m2| is the unit staggered magnetization with the AFM sublattice 
+magnetizations m1 and m2, ε is the staggered spin angular momentum density, A0 and A are 
+the homogeneous and inhomogeneous exchange constants, respectively, and L is the 
+parity-breaking constant [47]. For convenience of analytic calculations, one considers the 
+following approximation of the skyrmion configuration [48] in cylindrical coordinates with n 
+= (sin cos, sin sin, cos),  
+)
+cos
+cos
+arccos(
+|
+sin
+sin
+|
+sin
+sin
+2
+],
+)
+/
+sinh(
+)
+/
+sinh(
+arctan[
+2
+,
+)
+sin
+sin
+(
+)
+cos
+cos
+(
+0
+0
+0
+2
+0
+2
+0
+r
+R
+R
+R
+w
+r
+w
+R
+R
+R
+r
+s
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+−
+−
++
+=
+=
+−
++
+−
+=
+,
+ 
+(5) 
+where R is the skyrmion orbit radius, 0 is the polar angle of the skyrmion center, w = πD/4K 
+is the domain wall width, and Rs = πD[A/(16AK2 − π2D2K)]1/2 is the skyrmion radius. Then, 
+the equation describing the motion of skyrmion is obtained, 
+
+F
+q
+q
+=
++
+)
+(
+0
+
+
+
+
+A
+M
+, 
+ (6) 
+where M is the skyrmion effective mass, and q is the skyrmion coordinate [47]. The tangential 
+force F contains two terms, 
+V
+t
+m
+t
+m
+e
+W
+W
+A
+m
+LB
+F
+V
+t
+m
+t
+m
+e
+W
+W
+A
+B
+F
+W
+m
+m
+W
+m
+d
+)]
+sin
+sin
+cos
+)(cos
+cos(
+)
+cos
+sin
+sin
+)(cos
+sin(
+[
+)
+/
+(
+d
+)]
+cos
+cos
+sin
+)(sin
+cos(
+)
+sin
+cos
+sin
+)(cos
+[sin(
+)
+/
+(
+2
+/
+|
+|
+0
+0
+/
+|
+|
+0
+0
+2
+2
+2
+2
+2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+
+
+
++
+
+
+
++
+
+
+
++
+−
+=
+
+
++
+
+
+
+
+
+−
+
+
+=
+
+
+−
+−
+
+
+
+
+, 
+ (7) 
+where the sign + (−) in ± corresponds to the left- (right-) handed light. Here, F and Fm come 
+from the temporal and spatial deflection of the beam profile, respectively. A phase  term is 
+introduced in Fm to ensure that the two forces are always asynchronous. 
+In Fig. 3(a), we present the LLG-simulated trajectories of the AFM skyrmion driven by 
+left-handed light for m = −6, B0 = 1.1 T, W = 25a, σ = ∞, and  = 0.6 THz. Like the skyrmion 
+in ferromagnets, the AFM skyrmion is driven by the optical vortex rotating around the core 
+with a very high angular frequency under the restraint of the optical potential well [33]. Here, 
+the skyrmion rotates clockwise accompanying with an oscillation and breathing modes. 
+Furthermore, a reversed m generates an opposite OAM of the light, resulting in an 
+anticlockwise rotation of the skyrmion, as shown in Fig. 3(b), where the skyrmion trajectory 
+for m = 6 is presented. Besides the rotation direction, both the orbit radius and speed are 
+significantly changed. More interestingly, the rotation direction can also be controlled by 
+modulating the frequency . For example, for m = −6, an anticlockwise rotation is realized 
+when  is increased to 1.5 THz, as shown in Fig. 3(c).    
+To further explore this attractive physical phenomenon, we present the average tangential 
+velocity of the skyrmion v as a function of  for various m in Fig. 4(a). Similarly, there is a 
+starting frequency  ~ 0.4 THz beyond which the skyrmion can be effectively driven. Then, 
+the rotation speed first increases and then decreases with the increasing frequency, and it 
+reaches a maximum value ~900 m/s around  = 0.5 THz, which is much faster than that of 
+ferromagnetic one [33]. Importantly, there exists a critical frequency c ~ 1.0 THz, above 
+
+which the rotation direction is changed from clockwise to anticlockwise due to the reverse of 
+the OAM, which is different from the case of ferromagnets.  
+In some extent, the reverse of the rotation direction and OAM in antiferromagnets can be 
+understood qualitatively from the competition between F and Fm. For  = 0.6 THz which is 
+lower than c, the magnitude of Fm is larger than that of F as shown in Fig. 4(c), where the 
+calculated F, Fm, and Ft = F + Fm as functions of time are presented, resulting in the 
+clockwise rotation of the AFM skyrmion. The magnitude of F increases with the increase of 
+, while that of Fm hardly be affected by . Thus, when the two forces cancel each other out 
+at  ~ 1.0 THz as shown in Fig. 4(d), the skyrmion rotation is completely suppressed. For  > 
+c, F overwhelms Fm as shown in Fig. 4(e), resulting in an opposite OAM and 
+counter-rotation of the skyrmion. A further increase of  from 1.5 THz speeds down the 
+skyrmion gradually to zero, due to the mismatch between the optical frequency and the AFM 
+intrinsic frequency. 
+The critical frequency c depends on both the quantum number m and the orbit radius R, 
+which can be analytically estimated from Eq. (7). The calculated and simulated c as 
+functions of m are given in Fig. 4(b), which coincide well with each other. Interestingly, above 
+c, the rotation speed of the skyrmion significantly depends on the sign of m. For example, 
+the speed for m = 6 is one order of magnitude lower than that for m = −6, as shown in Fig. 
+4(a). It is noted that the OAM transfer from beam to the skyrmion is significantly determined 
+by the coupling of the OAM and local magnetization precession related to the beam chirality, 
+especially for high frequency. Thus, in the case of left-handed light for m = −6, the OAM and 
+magnetization precession are with a same direction, resulting in a strong OAM transfer. 
+Conversely, the OAM transfer is extensively suppressed for m = 6 due to their opposite 
+directions. Thus, the magnitude of Ft for m = −6 is much larger than that for m = 6, as shown 
+in Fig. 4(f), as well as the rotation speed of the skyrmion.   
+As a matter of fact, the above analysis can be easily transferred to the skyrmion dynamics 
+driven by right-handed light from symmetry analysis. Specifically, the right-handed light 
+coupling with positive m generates a force Ft which is opposite to that of the left-handed light 
+coupling with −m, as clearly shown in Fig. 4(f), where Ft induced by right-handed light for m 
+
+= 6 is also presented. As a result, the skyrmions for the two cases rotate reversely with a same 
+rotation speed.  
+Solving Eq. (5), the skyrmion velocity can be evaluated to be 
+)
+(
+2
+2
+2
+2
+0
+0
+0
+1
+3
+2
+)
+/
+(
+c
+C
+C
+A
+e
+A
+W
+MA
+F
+B
+C
+v
+
+
+
+
+
+
+
+
+−
+−
++
+=
+, 
+ (8) 
+where FA is the amplitude of Ft, C1 = 4.8, C2 = 2, and C3 = 2 are the fitting coefficients 
+estimated from the LLG-simulated results in Fig. 4(a). 
+Furthermore, the dependences of the skyrmion velocity on several physical parameters 
+including α, B0, W and m are investigated, and the corresponding results are shown in Fig. 5. 
+The Eq. (8) calculated (solid lines) and LLG-simulated (solid symbols) velocities for vairous 
+field B0, damping constant α, and beam waist W driven by the left-handed beam for m = −6 
+and right-handed beam for m = 6 are plotted in Figs. 5(a)-5(c), respectively. The simulated 
+results can be well fitted by Eq. (8), confirming the validity of above analysis. First, v  B02 is 
+obtained because the magnetic energy density linearly depends on B02. Second, an enhanced 
+damping term always reduces the skyrmion mobility, and the velocity linearly increases with 
+1/α. Furthermore, as W increasing, the rotation radius R is increased, while the beam energy 
+density is decreased, speeding down the skyrmion. Fig. 5(d) shows the velocity for various m, 
+which demonstrates the decreasing of v with the decreasing |m|, well consistent with Eq. (8).  
+For the integrity of this work, we also investigated the interaction between the AFM 
+skyrmion and multipolar spin waves which are excited by the vortex beam. Here, we consider 
+a high beam energy density and reset the parameters to be W = 5a and  = 3.5 THz to excite 
+spin waves, and we control the beam focus in a position far away from the skyrmion. The spin 
+wave mode has the following cylindrical wave form, 
+)]
+(
+exp[
+1
+t
+m
+k
+i
+
+
+
+
+−
++
+=
+
+, 
+ (9) 
+where k is the wave number, whose direction depending on the  angle. Then, the scattering 
+amplitude of the spin wave is given by [49]  
+/4
+/2
+2
+[(
+)
+/2]
+( )
+(1
+)
+,
+2
+l
+i
+il
+i
+i m l
+l
+l
+l
+m
+e
+e
+f
+e
+e
+m
+l
+k
+
+
+
+
+
+
+
+−
+−
+
+−
++
++
+=−
+−
+=
+−
++
+
+ 
+ (10) 
+
+where δl is the phase shift with the partial waves index l, and  is the scattering angle related 
+to the wave vector.  
+One notes that the OAM quantum number m is coupled with l and the phase shift, thus it 
+affects the scattering amplitude distribution and breaks the degeneracy between the 
+left-handed and right-handed magnons. The assumption is verified by the simulations, as 
+shown in Fig. 6 (a), where presents the skyrmion trajectories driven by the multipolar 
+right-handed (solid lines) and left-handed (dashed lines) spin waves for various m. The 
+multipolar spin waves drive the skyrmion propagation almost linearly, while the wave 
+handedness significantly affects the direction of the motion. Moreover, the obvious 
+asymmetry between the solid line and the dashed line for a same m with respect to the wave 
+vector direction (along the x-axis) demonstrates the degeneracy between the left-handed and 
+right-handed spin waves is broken, different from the case of plane spin waves [42,50]. 
+Importantly, the parameter m also affects the Hall motion of the skyrmion. In Fig. 6(b), we 
+give the dependence of Hall angle Hall = vy/vx on m, which demonstrates the suppression of 
+the Hall motion with the increase of |m|. Thus, it is suggested that the OAM quantum number 
+can effectively modulate the skyrmion Hall motion. 
+During the past of a few years, AFM skyrmions have been observed experimentally 
+[34,45], which are suggested to be information carriers in future spintronic devices for their 
+merits. Thus, searching for reliable methods in ultrafast manipulating AFM skyrmion is 
+extremely important for spintronic applications. In this work, the ultrafast generating/erasing 
+and dynamics of the AFM skyrmion by optical vortex are revealed using numerical 
+simulations, which is a certainly important step for spintronic device design based on AFM 
+skyrmion. Moreover, besides the OAM quantum number, the light frequency can also control 
+the direction of the skyrmion rotation, which allows one to modulate the skyrmion dynamics 
+easily through tuning the frequency. Furthermore, the skyrmion Hall motion driven by 
+multipolar spin waves excited by optical vortex can be controlled by modulating the light 
+handedness and OAM quantum number. One notes that the magnetic parameters considered 
+in this work are comparable with those in the antiferromagnets KMnF3 [36,51], and the 
+optical vortex can be achieved through spiral phase plates [10]. Thus, the prediction given 
+
+here does provide critical information for optical control AFM skyrmions, which deserves to 
+be checked in further experiments.  
+In conclusion, we have studied numerically and analytically the skyrmion dynamics in 
+antiferromagnets driven by optical vortex. The vortex beam excites twisted magnons and 
+generates/erases the skyrmion in a very short time of ~ps. The OAM transfer from the light 
+results in the rotation of the skyrmion, whose direction can be modulated by the light 
+frequency in addition to the OAM quantum number. This interesting behavior allows one to 
+modulate the skyrmion rotation easily through tuning the frequency. Furthermore, the optical 
+vortex excites multipolar spin waves, which in turn drives the skyrmion Hall motion. The Hall 
+angle also depends on the OAM quantum number, providing a new degree of freedom to 
+better control the skyrmion motion.  
+ 
+ 
+Acknowledgment 
+This work is supported by the Natural Science Foundation of China (Grants No. U22A20117, 
+No. 51971096, No. 92163210, and No. 51721001), the Guangdong Basic and Applied Basic 
+Research Foundation (Grant No. 2022A1515011727), and Funding by Science and 
+Technology Projects in Guangzhou (Grant No. 202201000008).  
+ 
+ 
+
+ 
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+Adv. Funct. Mater. 32, 2111906 (2022). 
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+[51] K. Saiki, J. Phys. Soc. Japan. 33, 1284 (1972). 
+ 
+ 
+
+ 
+FIGURE CAPTIONS 
+ 
+ 
+ 
+Fig. 1. The coupling principle of optical vortex and magnet, where the red ball represents a photon, and the 
+red and purple arrows represent the SAM and OAM, respectively. 
+ 
+
+optical vortex
+m=2
+X 
+ 
+ 
+Fig. 2. Ultrafast generation (a) and erasure (b) of isolated AFM skyrmion through optical vortex focusing. 
+ 
+
+(a)
+t=2ps
+t=20ps
+t=4ps
+t=6ps
+m=-8
+0.8
+0.6
+0.4
+0.2
+(b)
+0
+t=2ps
+t=3ps
+t=4ps
+t=6ps
+-0.2
+-0.4
+m=8
+0.6
+0.8 
+ 
+Fig. 3. Skyrmion gyration driven by optical vortex with various OAM quantum number m and light 
+frequency , (a) m = −6, and  = 0.6 THz, (b) m = 6, and  = 0.6 THz, and (c) m = −6, and  = 1.5 THz. 
+The white dotted circles are the trajectories of the skyrmion, whose radius R depends on both m and . 
+ 
+
+(a)
+25ps
+=
+50ps
+75ps
+100ps
+nz
+m= -6
+ZHL90=0
+0.8
+0.6
+(b)
+0.4
+451
+t=90ps
+=135ps
+ps
+180
+ps
+0.2
+m=6
+0
+0 = 0.6 THz
+0.2
+-0.4
+(c)
+t=110ps
+1=220ps
+t=330ps
+440ps
+-0.6
+m=-6
+-0.8
+@ = 1.5 THz 
+Fig. 4. (a) The dependence of skyrmion tangential velocity v on the OAM quantum number m and 
+frequency , and (b) the numerical simulated (blue solid squares) and analytical calculated (blue line) 
+critical frequency c for various m. Skyrmion orbit radius R are also presented by red stars. (c)-(e) 
+Analytically calculated tangential forces F , Fm and Ft for various , respectively, and (f) Ft for various 
+light handedness and m with  = 1.5 THz. 
+ 
+
+900
+left-handedm=-6
+1.4
+left-handed
+20
+600
+-left-handedm=6
+米
+1.2
+300
+(THz)
+米
+15.
+(nm)
+(s/w)
+米
+0
+R
+米
+10
+-300
+米
+0.8
+-600
+米
+0.6
+5
+(a)
+*
+-900
+米
+(b)
+0.5
+1.0
+1.5
+2.0
+2.5
+-8
+9-
+-4
+-2
+0
+2
+4
+6
+8
+の(THz)
+m
+0.4
+ZHL90=0
+0.4/0=1.0THz
+0.6
+1.5THz
+0.2
+1.5THz
+left-handed
+0.2
+0.2
+0.4
+0.1
+0.2
+00T
+0.0
+E0.0
+F
+0.0
+-hande
+-0.2
+0.2
+-0.2
+-0.4
+-0.1
+(c)
+0.4
+right-huided
+040
+(d)
+-0.6/(e)
+m=6
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+I (ps)
+t (ps)
+t (ps)
+t (ps) 
+ 
+Fig. 5. The simulated (solid squares) and analytically fitted (sold lines) v as functions of (a) the beam 
+intensity B0, (b) the damping coefficient , (c) the beam waist W and (d) the OAM quantum number m, for 
+left-handed light with m = -6 and right-handed light with m = 6.  
+ 
+
+600
+900
+left-handedm=-6
+right-handed m = 6
+400
+600
+300
+200
+S
+(m/
+(m/
+0
+0
+-300
+-200
+-600
+-400
+-900
+(a)
+(b)
+-600
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+0
+50
+100
+150
+200
+250
+B,(T)
+1/α
+1200
+400
+800
+200
+400
+(m/s)
+(s/u)
+0
+-400
+-200
+-800
+(c)
+(d)
+-400
+-1200
+20
+25
+30
+35
+40
+45
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+W
+m 
+ 
+Fig. 6. (a) The skyrmion trajectories driven by multipolar right-handed (solid lines) and left-handed (dashed 
+lines) spin waves induced by the optical vortex with various m, and (b) the dependence of the skyrmion 
+Hall angle Hall = vy/vx on m. The horizontal gray dashed line in (a) represents the wave vector direction. 
+
+6
+4
+2
+0
+1
+5
+5
+5
+4
+3
+3
+2
+2
+1
+Hall
+0
+09
+4
+-
+3
+5
+-
+-
+5
+1
+3
+-
+-
+1
+1
+3
+3
+-
+I
+-
+330
+1
+3
+-
+1
+X
+1
+2
+1
+3
+3
+(a)
+3
+1
+1
+5
+0
+0
+0
+2
+5
+3
+3
+2
+1

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